#144446

order	512
nilpotency class	4
isoclinism	isoclinic class of #142232
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

645 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 645 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=di,e^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,e^b=eh,e^c=eg,g^c=gi

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, 51642);