#137384

order	512
nilpotency class	4
isoclinism	isoclinic class of #137383
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

575 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44580);