#495124

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1306 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=f,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=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, 402320);