#509843

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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 ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

879 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 879 conjugacy classes of subgroups — too many to draw. Only its 144 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)(9,10,13,19,16,11,22,18,15,21,28,35,14,20,27,34)(12,17,24,31,29,23,36,30,26,33,38,40,25,32,37,39);(5,6)(7,8)(9,11)(10,16)(12,23)(13,19)(14,21)(15,20)(17,29)(18,27)(22,34)(24,31)(25,33)(26,32)(28,35)(30,37)(36,39)(38,40);(2,4)(6,8)(9,12,15,26)(10,17,21,33)(11,23,20,32)(13,24,28,38)(14,25,16,29)(18,30,34,39)(19,31,35,40)(22,36,27,37);(2,4)(10,18)(11,19)(13,27)(14,16)(17,30)(20,35)(21,34)(22,28)(23,31)(24,37)(25,29)(32,40)(33,39)(36,38)

Polycyclic presentation:

2^9|a^2=ef,c^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417039);