#72232001

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	C₂×C₄	#2177
commutator	C₁₆	C₂³×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
socle	C₂²	C₂×#2177
prime radical	C₂×#508	C₂×C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×#1311 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^10|a^2=hj,c^2=egj,d^2=f,e^2=gi,f^2=h,g^2=ij,i^2=j,c^a=ce,c^b=cj,d^c=di,e^a=eg,e^d=ej,f^c=fj,g^a=gi,i^a=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232005, #72234355, #72260494
maximal normal subgroup of	#72232005, #72234355, #72260494

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.10);;
gap> SetPower(coll, 3, F.5*F.7*F.10);;
gap> SetPower(coll, 4, F.6);;
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, 9, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, 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);