#499068

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

1196 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)(6,7)(9,10)(11,15)(12,16)(13,27)(14,17)(18,24)(19,25)(20,30)(21,34)(22,35)(23,29)(26,32)(28,31)(33,38)(36,40)(37,39);(5,6)(7,8)(9,11,14,23)(10,15,17,29)(12,20,26,37)(13,28,25,22)(16,30,32,39)(18,34,33,40)(19,35,27,31)(21,38,36,24);(2,4)(5,7,8,6)(10,16)(11,20)(12,24)(13,25)(17,32)(18,19)(21,22)(23,37)(26,38)(27,33)(28,36)(30,34)(31,35)(39,40);(10,17)(15,29)(16,32)(18,33)(19,27)(30,39)(31,35)(34,40)

Polycyclic presentation:

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

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, 406264);