#495875

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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

700 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)(9,10,13,17)(11,19,21,34)(12,24,26,30)(14,28,25,18)(15,29,31,22)(16,20,33,27)(23,40,38,36)(32,39,35,37);(9,11,20,37,14,23,12,22,13,21,27,39,25,38,26,29)(10,15,30,40,18,32,16,19,17,31,24,36,28,35,33,34);(2,4)(11,21)(15,31)(19,34)(22,29)(23,38)(32,35)(36,40)(37,39);(5,7,8,6)(11,21)(15,31)(19,34)(22,29)(23,38)(32,35)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=ehi,d^2=gi,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,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, 403071);