#503295

order	512
nilpotency class	4
isoclinism	isoclinic class of #501060
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

721 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 721 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,g^c=gi

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, 410491);