#7624979

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624802
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 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1411 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532175);