#137573

order	512
nilpotency class	4
isoclinism	isoclinic class of #137493
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#449
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

342 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=gi,e^2=ghi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=eg,f^b=fg,d^c=dg,e^c=egh,f^c=fg,h^c=hi

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, 44769);