#504422

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

615 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)(3,6)(4,5)(7,8)(9,10,13,19,15,20,30,14)(11,21,24,33,25,36,29,16)(12,17,26,34,28,35,27,18)(22,23,32,39,38,40,37,31);(9,11)(10,16)(12,23)(13,29)(14,21)(15,25)(17,22)(18,32)(19,36)(20,33)(24,30)(26,31)(27,39)(28,40)(34,37)(35,38);(2,5)(6,8)(9,12)(10,17)(11,22)(13,26)(14,18)(15,28)(16,31)(19,34)(20,35)(21,23)(24,32)(25,38)(27,30)(29,37)(33,39)(36,40);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,23)(13,27)(14,17)(15,28)(16,32)(19,35)(20,34)(21,22)(24,31)(25,40)(26,30)(29,39)(33,37)(36,38)

Polycyclic presentation:

2^9|a^2=e,d^2=f,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,g^b=gh,h^b=hi,g^c=gh,h^c=hi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411618);