#498566

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
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	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

696 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)(5,6,8,7)(9,10,13,19,27,35,24,16,15,21,12,18,14,20,28,30)(11,17,25,33,36,40,37,29,23,31,22,32,26,34,38,39);(5,7,8,6)(10,16)(11,22)(12,24)(13,28)(14,27)(18,20)(19,35)(21,30)(23,25)(26,38)(29,32)(33,39)(34,40)(36,37);(1,2,3,4)(11,23)(17,31)(22,25)(26,36)(29,39)(32,33)(34,40)(37,38);(9,11,24,37,14,26,13,25,15,23,28,38,27,36,12,22)(10,17,16,29,20,34,19,33,21,31,30,39,35,40,18,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=f,d^2=eh,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405762);