#72287435

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#106320
commutator	C₂×M₁₆	C₂×C₄×C₈
Frattini, cosocle	#33326	C₂³
socle	C₂	#72285965
prime radical	#3444	M₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #346 ◁ #33326 ◁ G
LECS	G ▷ #33326 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^11|b^2=d,c^2=efijk,d^2=h,e^2=jk,f^2=i,i^2=j,j^2=k,c^a=cg,c^b=ce,d^c=dj,e^a=eh,f^a=fhk,f^b=fj,f^d=fk,g^b=ghk,g^c=gk,g^e=gk,g^f=gk,h^c=hk,i^b=ik

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287480, #72319914
maximal normal subgroup of	#72287480, #72319914

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