#6516926

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500157
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3314 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:

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

Polycyclic presentation:

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

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, 6424122);