#499898

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1556 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1556 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407094);