#351021

order	512
nilpotency class	3
isoclinism	isoclinic class of #348318
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

586 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fhi,c^2=ei,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258217);