#511952

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
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×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 970 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)(9,10,13,16)(11,19,22,33)(12,24,26,39)(14,27,25,18)(15,28,29,21)(17,32,31,20)(23,38,37,35)(30,36,34,40);(9,11,20,36,14,23,26,28,13,22,32,40,25,37,12,21)(10,15,24,35,18,30,31,33,16,29,39,38,27,34,17,19);(2,4)(9,10)(11,15)(12,17)(13,16)(14,18)(19,21)(20,24)(22,29)(23,30)(25,27)(26,31)(28,33)(32,39)(34,37)(35,36)(38,40);(6,8)(10,16)(15,29)(17,31)(18,27)(19,33)(24,39)(30,34)(35,38)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,h^a=hi,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, 419148);