#72232018

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#357
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	#7623249	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ C₄×#504 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1075 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:

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

Polycyclic presentation:

2^10|a^2=hi,b^2=hi,d^2=efhj,e^2=gi,f^2=h,g^2=ij,h^2=j,i^2=j,b^a=bhi,c^b=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

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232026, #72322316, #72322477
maximal normal subgroup of	#72232026, #72322316, #72322477

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, 2, 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, 2, 1, F.2*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, 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);