#512293

order	512
nilpotency class	4
isoclinism	isoclinic class of #511887
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=ef,d^2=e,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,g^d=ghi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419489);