#494288

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

1100 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=i,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,e^b=eh,f^b=fh,h^b=hi,e^c=eh,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, 401484);