#6516939

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500006
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3245 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,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, 6424135);