#493766

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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

1130 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)(9,10)(11,16)(12,18)(13,26)(14,30)(15,20)(17,22)(19,28)(21,31)(23,34)(24,33)(25,37)(27,32)(29,40)(35,39)(36,38);(1,2,3,4)(9,11)(10,16)(12,24)(13,22)(14,21)(15,23)(17,26)(18,33)(19,32)(20,34)(25,39)(27,28)(29,38)(30,31)(35,37)(36,40);(9,11)(10,17)(12,21)(13,27)(14,24)(15,23)(16,26)(18,32)(19,33)(20,36)(22,28)(25,38)(29,39)(30,35)(31,37)(34,40);(9,12,15,14)(10,18,20,30)(11,21,23,24)(13,25,29,28)(16,31,34,33)(17,32,36,35)(19,26,37,40)(22,27,38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400962);