#7624895

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 72 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532091);