#135352

order	512
nilpotency class	4
isoclinism	isoclinic class of #124849
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

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

335 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

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

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, 42548);