#499169

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³×D₁₆	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1174 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1174 conjugacy classes of subgroups — too many to draw. Only its 234 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)(11,21)(12,16)(13,18)(14,29)(15,20)(17,22)(19,26)(23,35)(24,32)(25,34)(27,31)(28,30)(33,37)(36,39)(38,40);(5,6)(7,8)(10,16)(11,22)(12,13)(14,26)(17,32)(18,19)(20,31)(23,37)(24,25)(27,28)(29,30)(33,40)(34,36)(38,39);(2,4)(6,8)(9,11,12,24,26,39,28,37,15,23,27,38,14,25,13,22)(10,17,18,34,29,40,31,35,20,33,30,36,19,32,16,21);(11,23)(17,33)(21,35)(22,37)(24,38)(25,39)(32,40)(34,36)

Polycyclic presentation:

2^9|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, 406365);