#495218

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×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1182 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)(10,15)(11,20)(12,16)(13,23)(17,31)(18,34)(19,28)(21,35)(22,37)(24,30)(26,27)(29,36)(32,33)(39,40);(1,2,3,4)(6,8)(9,10,16,27,13,18,24,28,14,19,30,34,23,26,12,15)(11,17,29,39,22,33,35,38,21,32,37,40,36,31,20,25);(9,11,16,29,13,22,24,35,14,21,30,37,23,36,12,20)(10,17,27,39,18,33,28,38,19,32,34,40,26,31,15,25);(11,21)(17,32)(20,35)(22,36)(25,38)(29,37)(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^c=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, 402414);