#503430

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61084	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

985 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 985 conjugacy classes of subgroups — too many to draw. Only its 156 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)(3,7)(4,6)(5,8)(9,10,15,20)(11,21,25,39)(12,26,28,18)(13,17,30,36)(14,31,27,19)(16,32,35,23)(22,34,29,38)(24,37,40,33);(1,3,5,4)(2,6,8,7)(11,22)(12,13)(14,27)(16,33)(17,26)(18,36)(19,31)(21,34)(23,40)(24,32)(25,29)(28,30)(35,37)(38,39);(1,3,5,4)(2,7,8,6)(9,11,13,29,14,24,28,32,15,25,30,22,27,40,12,23)(10,16,26,33,19,34,36,39,20,35,18,37,31,38,17,21);(1,4,5,3)(2,7,8,6)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=efh,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, 410626);