#7623138

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622825
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 ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,15,13)(11,14,19,17)(12,23,26,38)(16,31,30,22)(18,35,24,27)(20,29,28,33)(21,32,36,39)(25,40,34,37);(2,5)(6,8)(9,11)(10,17)(12,21)(13,14)(15,19)(16,22)(18,34)(20,33)(23,39)(24,25)(26,36)(27,37)(28,29)(30,31)(32,38)(35,40);(9,12,15,26)(10,18,13,24)(11,21,19,36)(14,25,17,34)(16,32,30,39)(20,37,28,40)(22,38,31,23)(27,29,35,33);(1,3,4,7)(2,6,5,8)(9,13,15,10)(11,14,19,17)(12,24,26,18)(16,28,30,20)(21,25,36,34)(22,29,31,33)(23,27,38,35)(32,40,39,37);(9,14,15,17)(10,19,13,11)(12,25,26,34)(16,29,30,33)(18,36,24,21)(20,31,28,22)(23,40,38,37)(27,32,35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530334);