#145726

order	512
nilpotency class	4
isoclinism	isoclinic class of #145724
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

322 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

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, 52922);