#136786

order	512
nilpotency class	3
isoclinism	isoclinic class of #9443
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂³	C₂×C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 330 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) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egi,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,f^c=fgi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43982);