#7611874

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563346
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

1921 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)(13,28)(14,31)(15,18)(16,21)(17,23)(19,30)(24,39)(25,35)(26,34)(27,36)(29,32)(33,40)(37,38);(1,3)(2,5)(4,7)(6,8)(9,11,15,23,16,25,12,26)(10,17,20,22,21,34,18,35)(13,29,14,24,30,38,27,40)(19,33,31,37,28,39,36,32);(2,6)(3,7)(9,12,16,15)(10,18,21,20)(11,23,25,26)(13,27,30,14)(17,22,34,35)(19,36,28,31)(24,38,40,29)(32,33,37,39);(2,6)(5,8)(9,13,15,14,16,30,12,27)(10,19,18,36,21,28,20,31)(11,24,26,29,25,40,23,38)(17,32,22,33,34,37,35,39);(9,14,15,30,16,27,12,13)(10,19,20,31,21,28,18,36)(11,24,23,38,25,40,26,29)(17,33,22,37,34,39,35,32)

Polycyclic presentation:

2^9|b^2=f,c^2=i,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,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, 7519070);