#72322977

order	2048
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72322878
commutator, Frattini, cosocle	C₂²×C₃₂	C₂⁴
prime radical	#72322440	C₂
LCS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2929 ◁ #33432 ◁ #512742 ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 128 points:

(1,2,5,11,29,45,86,105,8,13,32,47,40,52,80,85,36,4,23,25,74,81,107,124,94,27,76,82,7,12,31,46,6,15,30,49,39,51,91,92,35,54,89,108,95,26,75,28,9,14,33,48,87,37,96,84,38,50,90,106,34,53,88,93)(3,16,19,55,66,103,118,127,21,57,68,110,73,78,115,83,60,18,61,62,112,97,128,126,104,64,114,116,20,56,67,109,17,10,41,42,72,100,121,122,59,44,99,102,111,63,113,65,22,24,69,77,119,70,123,117,71,79,120,125,58,43,98,101);(1,3)(2,10)(4,24)(5,19)(6,17)(7,20)(8,21)(9,22)(11,42)(12,43)(13,44)(14,18)(15,16)(23,61)(25,77)(26,78)(27,79)(28,83)(29,66)(30,41)(31,67)(32,68)(33,69)(34,58)(35,59)(36,60)(37,97)(38,71)(39,72)(40,73)(45,100)(46,101)(47,102)(48,62)(49,55)(50,64)(51,103)(52,63)(53,56)(54,57)(65,85)(70,81)(74,112)(75,113)(76,114)(80,115)(82,125)(84,126)(86,118)(87,119)(88,98)(89,99)(90,120)(91,121)(92,127)(93,109)(94,104)(95,111)(96,123)(105,122)(106,116)(107,128)(108,110)(117,124);(3,17)(19,41)(20,58)(21,59)(22,60)(61,69)(66,72)(67,98)(68,99)(71,104)(73,111)(112,119)(113,115)(114,120)(118,121)(123,128);(1,4,9,15,6,14,36,2)(3,18,22,10,17,24,60,16)(5,28,33,93,30,85,23,46)(7,37,40,45,34,81,95,51)(8,13,38,27,35,54,94,50)(11,31,25,80,49,88,48,75)(12,39,26,74,53,29,52,87)(19,65,69,101,41,83,61,109)(20,70,73,103,58,97,111,100)(21,57,71,64,59,44,104,79)(32,92,90,84,89,105,76,124)(42,98,77,113,55,67,62,115)(43,66,78,119,56,72,63,112)(47,107,82,86,108,96,106,91)(68,122,120,117,99,127,114,126)(102,123,125,121,110,128,116,118)

Polycyclic presentation:

2^11|a^2=f,d^2=gjk,f^2=hik,h^2=ij,i^2=jk,j^2=k,b^a=bk,c^a=ce,c^b=cg,d^a=dfjk,d^b=dk,d^c=de,e^b=ek,f^d=fh,g^a=gk,g^d=gk,h^d=hi,i^d=ij,j^d=jk

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322999
maximal normal subgroup of	#72322999

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 4, F.7*F.10*F.11);;
gap> SetPower(coll, 6, F.8*F.9*F.11);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetPower(coll, 9, F.10*F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 4, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);