#6352111

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352061
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3000 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259307);