#72227006

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#1380
commutator	C₂×C₁₆	C₄×C₁₆
Frattini, cosocle	C₂×C₁₆²	C₂²
socle	C₂²	#92931
prime radical	C₂³	#7068
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ C₂×C₈×C₁₆ ◁ C₂×C₁₆² ◁ G
LECS	G ▷ C₂×C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=e,b^2=c,c^2=f,d^2=g,f^2=h,g^2=i,h^2=j,i^2=k,b^a=bd,c^a=cg,d^a=dgijk,e^b=ej,f^a=fi,g^a=gik,h^a=hk,i^a=ik

Conjugacy classes & character table

368 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227028, #72227135, #72227142
maximal normal subgroup of	#72227028, #72227135, #72227142
maximal quotient of	#72227007

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