#493053

order	512
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂³×D₈	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=i,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, 400249);