#135350

order	512
nilpotency class	4
isoclinism	isoclinic class of #124679
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

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

normal subgroups only

This group has 330 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)(11,20,23,18)(12,24)(13,17)(14,19)(15,22,29,36)(16,21,31,35)(25,28)(26,33)(27,32)(30,34,39,40)(37,38);(9,11)(10,15,19,29)(12,21)(13,22)(14,23)(16,28,31,38)(17,20,32,18)(24,30,33,39)(25,34)(26,35)(27,36)(37,40);(1,2)(3,7)(4,6)(5,8)(9,12,13,25,14,26,27,37)(10,16,17,30,19,31,32,39)(11,21,22,34,23,35,36,40)(15,28,20,33,29,38,18,24)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gi,f^b=fi,e^c=eh,f^c=fgh,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, 42546);