#135325

order	512
nilpotency class	4
isoclinism	isoclinic class of #135322
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1823
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 526 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)(3,7,8,5)(4,6)(9,10,13,20,15,18,14,30)(11,21,23,34,24,16,29,33)(12,25,27,35,28,17,26,19)(22,31,38,37,39,36,40,32);(9,11,15,24)(10,16,18,21)(12,22,28,39)(13,29,14,23)(17,31,25,36)(19,32,35,37)(20,34,30,33)(26,38,27,40);(1,3)(2,5)(4,8)(6,7)(10,17,18,25)(12,26)(13,14)(16,31,21,36)(19,20,35,30)(22,38)(23,29)(27,28)(32,34,37,33)(39,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=d,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,h^a=hi,f^b=fi,e^c=eh,f^c=fgh,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, 42521);