#495703

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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

1150 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)(9,10,13,18)(11,16,20,30)(12,15,25,28)(14,26,24,19)(17,33,29,23)(21,27,34,38)(22,37,35,32)(31,40,39,36);(10,15)(12,23)(14,24)(16,27)(17,19)(18,28)(21,36)(22,35)(25,33)(26,29)(30,38)(31,32)(34,40)(37,39);(2,4)(9,11)(10,16)(12,21)(13,20)(14,22)(15,27)(17,31)(18,30)(19,32)(23,36)(24,35)(25,34)(26,37)(28,38)(29,39)(33,40);(5,7,8,6)(11,20)(16,30)(21,34)(22,35)(27,38)(31,39)(32,37)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402899);