#7611891

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563341
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=f,d^2=h,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7519087);