#135543

order	512
nilpotency class	4
isoclinism	isoclinic class of #135543
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8748
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

497 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42739);