#6432109

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431795
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	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3496 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)(11,15)(12,16)(13,17)(14,18)(19,28)(20,26)(21,27)(22,29)(23,30)(24,31)(25,33)(32,38)(34,37)(35,39)(36,40);(2,5)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,28)(17,26)(18,27)(22,32)(23,33)(24,34)(25,30)(29,38)(31,37)(35,40)(36,39);(9,12,14,23)(10,16,18,30)(11,19,21,33)(13,22,24,35)(15,25,27,28)(17,29,31,39)(20,32,34,40)(26,36,37,38);(1,3,4,7)(2,6,5,8)(13,24)(17,31)(20,34)(22,35)(26,37)(29,39)(32,40)(36,38);(9,13,14,24)(10,17,18,31)(11,20,21,34)(12,22,23,35)(15,26,27,37)(16,29,30,39)(19,32,33,40)(25,36,28,38)

Polycyclic presentation:

2^9|c^2=i,d^2=f,e^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,g^b=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, 6339305);