#511750

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#578	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3095 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 3095 conjugacy classes of subgroups — too many to draw. Only its 230 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)(11,19,21,38)(12,23,26,37)(14,28,25,18)(15,30,31,20)(17,34,33,27)(22,39,29,35)(24,36,40,32);(9,11,13,21)(10,15,16,31)(12,24,26,40)(14,29,25,22)(17,35,33,39)(18,36,28,32)(19,37,38,23)(20,34,30,27);(2,4)(9,10)(11,15)(12,17)(13,16)(14,18)(19,20)(21,31)(22,32)(23,27)(24,35)(25,28)(26,33)(29,36)(30,38)(34,37)(39,40);(6,8)(10,16)(15,31)(17,33)(18,28)(19,38)(23,37)(32,36)(35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418946);