#495213

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	C₂²×#508	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1126 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)(10,15)(11,20)(12,16)(13,23)(17,31)(18,26)(19,34)(21,37)(22,36)(24,30)(27,28)(29,35)(32,33)(39,40);(1,2,3,4)(6,8)(9,10,16,28,13,19,24,26,14,18,30,34,23,27,12,15)(11,17,29,39,21,33,36,38,22,32,37,40,35,31,20,25);(9,11,16,29,13,21,24,36,14,22,30,37,23,35,12,20)(10,17,28,39,19,33,26,38,18,32,34,40,27,31,15,25);(10,18)(15,26)(17,32)(19,27)(25,38)(28,34)(31,33)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,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, 402409);