#503432

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	#61086	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 943 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,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, 410628);