#7623756

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623626
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 ◁ #525 ◁ #2929 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1502 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530952);