#72232003

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#357
commutator	C₁₆	C₂³×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
socle	C₂²	C₂²×#357
prime radical	C₂²×#508	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ C₂²×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1423 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

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

Polycyclic presentation:

2^10|a^2=hi,d^2=efhj,e^2=gi,f^2=h,g^2=ij,h^2=j,i^2=j,c^a=cj,d^a=de,d^b=dj,e^a=eg,e^d=ej,f^a=fi,f^d=fj,g^a=gi,h^a=hj,i^a=ij

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232005
maximal normal subgroup of	#72232005

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