#6492418

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470818
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1825 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,g^c=gi,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, 6399614);