#72232012

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, socle	C₂²	C₂²×#367
commutator	C₁₆	C₂³×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	C₂×#41	#107
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ C₄×#504 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

665 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:

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

Polycyclic presentation:

2^10|a^2=f,d^2=egi,e^2=gij,f^2=h,g^2=ij,i^2=j,c^b=cj,d^a=de,e^a=eg,f^d=fj,g^a=gi,i^a=ij

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232013, #72321470, #72321484, …
maximal normal subgroup of	#72232013, #72321470, #72321484, …

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