#72225408

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#8266
commutator	C₄×C₈	C₂×C₄×C₈
Frattini, cosocle	C₄×M₆₄	C₂³
socle	C₂	#72225407
prime radical	C₂³	#13642
LCS	G ▷ C₄×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #7330 ◁ #149495 ◁ G
LECS	G ▷ C₄×M₆₄ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

404 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:

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

Polycyclic presentation:

2^11|a^2=d,b^2=e,c^2=fi,d^2=g,f^2=h,g^2=i,i^2=j,j^2=k,b^a=bf,c^a=cfhijk,c^b=ck,d^b=dj,e^a=eh,e^d=ek,f^a=fhjk,g^b=gk,h^a=hk

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225417, #72225510, #72225523, …
maximal normal subgroup of	#72225417, #72225510, #72225523, …

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