#72231905

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×C₄²
commutator	C₂×C₄	C₂²×C₄×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
socle	C₂²	C₂²×#346
prime radical	C₂×#41	C₄×C₈
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

719 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:

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

Polycyclic presentation:

2^10|a^2=e,c^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=j,c^a=chj,c^b=chij,d^b=dj,e^c=ej,f^a=fj

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231908, #72231969
maximal normal subgroup of	#72231908, #72231969

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