#503085

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
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	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

707 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 707 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fh,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,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, 410281);