#135389

order	512
nilpotency class	4
isoclinism	isoclinic class of #124679
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	#1820
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

354 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,5)(4,6)(7,8)(9,10)(11,21,14,19)(12,24)(13,18)(15,20)(16,23,32,29)(17,27,34,22)(25,31)(26,30)(28,35)(33,39,37,36)(38,40);(2,5)(6,7)(9,11,15,14)(10,16)(12,22,28,27)(13,23,25,29)(17,30)(18,21)(19,31)(20,32)(24,37)(26,36,38,39)(33,35)(34,40);(2,6)(4,8)(5,7)(9,12,25,38,15,28,13,26)(10,17,18,33,20,34,31,37)(11,22,29,39,14,27,23,36)(16,30,21,35,32,40,19,24)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42585);