#135347

order	512
nilpotency class	4
isoclinism	isoclinic class of #135346
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1824
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁵	Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 530 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)(9,10,13,17,14,19,29,18)(11,20,23,38,21,37,28,35)(12,24,27,16,25,36,40,34)(15,30,32,22,31,26,33,39);(12,25)(16,34)(20,37)(22,39)(24,36)(26,30)(27,40)(35,38);(1,3,4,8)(2,5,6,7)(9,11,13,28,14,21,29,23)(10,15,18,32,19,31,17,33)(12,26,27,22,25,30,40,39)(16,35,24,20,34,38,36,37)

Polycyclic presentation:

2^9|a^2=f,c^2=dg,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,h^a=hi,e^b=ei,d^c=di,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, 42543);