#136512

order	512
nilpotency class	4
isoclinism	isoclinic class of #136511
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8921
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

372 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=g,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dgi,e^c=ehi,f^c=fi,g^c=gi

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, 43708);