#495219

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×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1102 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)(12,21)(13,24)(16,25)(17,32)(18,30)(19,29)(22,35)(23,31)(27,28)(33,34)(36,37)(39,40);(1,2,3,4)(6,8)(9,10,16,30,13,18,12,15,14,19,21,28,24,27,25,29)(11,17,31,40,23,34,20,26,22,33,37,39,36,32,35,38);(9,11,21,37,13,23,25,35,14,22,16,31,24,36,12,20)(10,17,28,39,18,34,29,38,19,33,30,40,27,32,15,26);(11,22)(17,33)(20,35)(23,36)(26,38)(31,37)(32,34)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,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, 402415);