#137193

order	512
nilpotency class	4
isoclinism	isoclinic class of #137153
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
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 ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

539 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,7,8,5)(4,6)(9,10,14,19,15,21,30,33,16,22,12,28,13,20,29,17)(11,23,26,32,25,36,40,37,27,38,24,39,31,18,34,35);(1,2)(3,5)(4,6)(7,8)(10,17,22,33)(12,29)(13,15)(14,30)(18,32,36,39)(19,21,28,20)(23,35,38,37)(24,34)(25,31)(26,40);(1,3,4,8)(2,5,6,7)(9,11,16,27)(10,18,22,36)(12,26,14,24)(13,31,15,25)(17,32,33,39)(19,37,28,35)(20,38,21,23)(29,40,30,34)

Polycyclic presentation:

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

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, 44389);