#492858

order	512
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1642 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)(5,6,7,8)(10,16)(12,25)(13,27)(14,26)(17,28)(18,19)(20,31)(22,36)(23,35)(24,37)(29,30)(32,38)(33,34)(39,40);(2,4)(9,10,13,18,14,19,12,16,15,20,25,30,26,29,27,31)(11,17,23,33,24,34,22,28,21,32,36,40,37,39,35,38);(9,11)(10,17)(12,22)(13,23)(14,24)(15,21)(16,28)(18,33)(19,34)(20,32)(25,36)(26,37)(27,35)(29,39)(30,40)(31,38);(11,21)(17,32)(22,35)(23,36)(24,37)(28,38)(33,40)(34,39)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400054);