#502759

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

829 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 829 conjugacy classes of subgroups — too many to draw. Only its 150 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)(4,5)(6,8)(9,10,13,18)(11,20,22,17)(12,14,26,16)(15,19,28,31)(21,24,37,39)(23,25,33,34)(27,29,40,32)(30,38,36,35);(2,5)(6,7)(9,11)(10,17)(12,25)(13,22)(14,23)(15,30)(16,33)(18,20)(19,35)(21,27)(24,32)(26,34)(28,36)(29,39)(31,38)(37,40);(9,12,14,18,13,26,16,10)(11,21,23,38,22,37,33,35)(15,31,29,40,28,19,32,27)(17,30,25,39,20,36,34,24);(1,3,4,8)(2,6,5,7)(10,18)(12,26)(17,20)(19,31)(21,37)(25,34)(27,40)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409955);