#494863

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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

826 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,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, 402059);