#500696

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1115 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)(7,8)(9,10)(11,20)(12,17)(13,16)(14,28)(15,22)(18,27)(19,26)(21,36)(23,39)(24,31)(25,32)(29,35)(30,37)(33,40)(34,38);(1,2,4,3)(9,11,12,22)(10,15,17,20)(13,21,25,37)(14,29,26,24)(16,30,32,36)(18,34,33,39)(19,35,28,31)(23,40,38,27);(6,8)(10,16)(11,21)(13,27)(14,26)(17,32)(18,19)(22,37)(23,24)(25,40)(28,33)(29,38)(30,34)(31,35)(36,39);(1,3,4,2)(10,17)(15,20)(16,32)(18,33)(19,28)(30,36)(31,35)(34,39)

Polycyclic presentation:

2^9|b^2=e,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,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, 407892);