#6445206

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1947 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

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, 6352402);