#6432092

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431723
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 ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3362 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,18)(13,19)(14,27)(15,20)(17,24)(21,30)(22,31)(23,33)(25,35)(26,36)(28,32)(29,38)(34,37)(39,40);(2,5)(6,8)(9,11,15,23)(10,16,20,33)(12,21,25,39)(13,22,14,28)(17,34,36,29)(18,30,35,40)(19,31,27,32)(24,37,26,38);(9,12)(10,17)(11,21)(13,24)(14,26)(15,25)(16,29)(18,27)(19,35)(20,36)(22,37)(23,39)(28,38)(30,31)(32,40)(33,34);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,21)(13,26)(14,24)(15,25)(16,30)(17,27)(19,36)(20,35)(22,38)(23,39)(28,37)(29,31)(32,34)(33,40);(9,13,15,14)(10,19,20,27)(11,22,23,28)(12,24,25,26)(16,31,33,32)(17,35,36,18)(21,37,39,38)(29,30,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339288);