#72323065

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#12067
commutator	C₁₆	C₂³×C₁₆
Frattini, cosocle	#1311	C₂⁴
prime radical	C₂²×D₃₂	C₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #122214 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 80 points:

(1,2,4,8,5,9,11,15,6,10,14,3,7,12,16,13)(17,18,20,28,22,30,34,56,24,27,23,48,41,32,39,31)(19,26,36,57,38,59,68,60,35,61,49,78,71,62,33,67)(21,42,45,63,46,25,50,53,47,54,40,29,64,52,43,66)(37,73,75,58,44,77,69,80,70,51,72,55,74,79,76,65);(18,25)(19,33)(20,39)(21,43)(22,41)(23,34)(26,55)(27,52)(28,29)(30,42)(31,66)(32,54)(35,68)(36,71)(37,74)(38,49)(40,46)(44,70)(45,64)(47,50)(48,63)(51,78)(53,56)(57,73)(58,61)(59,80)(60,79)(62,65)(67,77)(72,75);(1,3,11,2,7,15,4,12,6,8,16,10,5,13,14,9)(17,19,34,68,41,71,20,36,24,35,39,33,22,38,23,49)(18,26,56,60,32,62,28,57,27,61,31,67,30,59,48,78)(21,44,50,72,64,37,45,69,47,74,43,75,46,70,40,76)(25,51,29,65,42,77,53,55,52,73,63,80,54,79,66,58);(18,27)(19,35)(25,52)(28,48)(29,63)(30,32)(31,56)(33,68)(36,49)(37,70)(38,71)(42,54)(44,74)(53,66)(69,76)(72,75)

Polycyclic presentation:

2^11|a^2=e,c^2=eg,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,b^a=bf,c^b=chjk,d^a=dk,d^c=dk,e^b=ejk,f^a=fh,f^b=fhjk,f^c=fj,f^e=fk,g^b=gk,h^a=hj,h^b=hjk,h^c=hk,j^a=jk,j^b=jk

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323084, #72323120, #72323152
maximal normal subgroup of	#72323084, #72323120, #72323152

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