#7611889

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

1915 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)(11,21)(12,26)(13,29)(14,20)(15,17)(16,24)(18,28)(19,31)(22,36)(23,39)(25,34)(27,32)(30,35)(33,38)(37,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,24,15,22,31,25)(10,16,19,21,17,34,20,36)(12,27,13,23,28,40,30,38)(18,33,29,37,26,39,35,32);(2,6)(3,7)(10,17)(11,22)(12,28)(13,30)(19,20)(24,25)(32,37)(33,39);(2,6)(5,8)(9,12,14,13,15,28,31,30)(10,18,20,35,17,26,19,29)(11,23,25,27,22,38,24,40)(16,32,21,33,34,37,36,39);(9,13,14,28,15,30,31,12)(10,18,19,29,17,26,20,35)(11,23,24,40,22,38,25,27)(16,33,21,37,34,39,36,32)

Polycyclic presentation:

2^9|b^2=f,d^2=h,e^2=f,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, 7519085);