#72232004

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#367
commutator	C₁₆	C₂³×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
socle	C₂²	C₂²×#367
prime radical	C₂⁴	#367
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

691 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:

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

Polycyclic presentation:

2^10|a^2=f,d^2=egi,e^2=gij,f^2=h,g^2=ij,i^2=j,c^a=cj,d^a=de,d^b=dj,e^a=eg,f^d=fj,g^a=gi,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.6);;
gap> SetPower(coll, 4, F.5*F.7*F.9);;
gap> SetPower(coll, 5, F.7*F.9*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*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, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);