#495217

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
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	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

1158 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fhi,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,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, 402413);