#499170

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1094 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,e^2=hi,f^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,h^a=hi,c^b=cg,e^b=eh,f^b=fhi,g^b=ghi,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406366);