#511957

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1028 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,h^a=hi,c^b=ci,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419153);