#144659

order	512
nilpotency class	4
isoclinism	isoclinic class of #144659
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10393
commutator, Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

344 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51855);