#494913

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,d^2=i,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402109);