#7623631

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623626
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

1923 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fh,h^a=hi,d^b=df,e^b=ei,f^b=fh,h^b=hi,f^d=fh,h^d=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, 7530827);