#508017

order	512
nilpotency class	4
isoclinism	isoclinic class of #508016
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 ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1023 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1023 conjugacy classes of subgroups — too many to draw. Only its 148 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)(3,6)(4,5)(7,8)(9,10,14,20)(11,15,24,30)(12,16,27,33)(13,18,26,19)(17,34,32,25)(21,35,23,29)(22,31,37,40)(28,38,39,36);(2,5)(6,8)(9,11)(10,15)(12,22)(13,21)(14,24)(16,31)(17,28)(18,35)(19,29)(20,30)(23,26)(25,36)(27,37)(32,39)(33,40)(34,38);(1,3)(2,6)(4,7)(5,8)(10,16)(11,21)(12,25)(13,26)(15,28)(17,19)(18,32)(20,33)(22,37)(23,24)(27,34)(29,40)(30,39)(31,35);(3,7)(6,8)

Polycyclic presentation:

2^9|a^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi,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, 415213);