#349415

order	512
nilpotency class	3
isoclinism	isoclinic class of #346882
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

797 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 797 conjugacy classes of subgroups — too many to draw. Only its 160 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=eh,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256611);