#72321587

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₈² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₈×C₂²⋊C₈
commutator	C₂×C₈	C₂×C₈²
Frattini, cosocle	C₂×#1397	C₂³
socle	C₂	#72286359
prime radical	C₂×#41	#346
LCS	G ▷ C₂×C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#7330 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 128 points:

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

Polycyclic presentation:

2^11|a^2=e,b^2=h,c^2=f,e^2=g,f^2=h,g^2=i,i^2=j,j^2=k,b^a=bd,c^a=cijk,d^b=dk,e^c=ej,f^a=fjk,f^e=fk,g^c=gk,h^a=hk

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321722, #72322125, #72322743
maximal normal subgroup of	#72321722, #72322125, #72322743

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.5);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);