#7624232

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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

1500 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)(9,10,14,21)(11,24,27,17)(12,19,30,47)(13,23,16,20)(15,36,35,51)(18,43,45,31)(22,34,52,38)(25,54,57,40)(26,41,32,49)(28,62,61,66)(29,50,33,46)(37,48,65,44)(39,58,53,71)(42,60,55,64)(56,69,59,70)(63,68,72,67);(1,3)(2,5)(4,8)(6,7)(9,11,14,27)(10,17,21,24)(12,25,30,57)(13,32,16,26)(15,28,35,61)(18,39,45,53)(19,40,47,54)(20,49,23,41)(22,42,52,55)(29,59,33,56)(31,58,43,71)(34,64,38,60)(36,66,51,62)(37,63,65,72)(44,68,48,67)(46,70,50,69);(9,12)(10,18)(11,25)(13,29)(14,30)(15,37)(16,33)(17,39)(19,36)(20,44)(21,45)(22,46)(23,48)(24,53)(26,56)(27,57)(28,63)(31,38)(32,59)(34,43)(35,65)(40,66)(41,67)(42,70)(47,51)(49,68)(50,52)(54,62)(55,69)(58,60)(61,72)(64,71);(3,8)(5,7)(9,12)(10,19)(11,25)(13,33)(14,30)(15,31)(16,29)(17,40)(18,36)(20,50)(21,47)(22,48)(23,46)(24,54)(26,59)(27,57)(28,58)(32,56)(34,65)(35,43)(37,38)(39,66)(41,70)(42,67)(44,52)(45,51)(49,69)(53,62)(55,68)(60,63)(61,71)(64,72);(2,6)(5,7)(9,13,14,16)(10,20,21,23)(11,26,27,32)(12,29,30,33)(15,34,35,38)(17,41,24,49)(18,44,45,48)(19,46,47,50)(22,51,52,36)(25,56,57,59)(28,60,61,64)(31,37,43,65)(39,67,53,68)(40,69,54,70)(42,66,55,62)(58,72,71,63)

Polycyclic presentation:

2^9|a^2=i,b^2=i,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=dfi,e^b=ei,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531428);