#135293

order	512
nilpotency class	4
isoclinism	isoclinic class of #124651
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁵	D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

570 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=d,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,h^a=hi,e^c=eh,f^c=fghi,g^c=gi,h^c=hi,f^e=fi

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, 42489);