#6491818

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470421
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

1898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,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, 6399014);