#137869

order	512
nilpotency class	3
isoclinism	isoclinic class of #1665
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	#3106	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

335 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,6,10,16)(4,9)(5,8)(7,11,15,12)(13,14)(17,18,21,26,23,25,36,38,22,27,35,41,20,34,29,44)(19,28,31,42,33,40,46,47,32,43,45,48,30,24,37,39);(1,3)(2,6,8,16)(4,12)(5,10)(7,14,15,9)(11,13)(17,19,29,37,20,30,35,45,22,32,36,46,23,33,21,31)(18,24,38,42,34,43,26,39,27,40,44,48,25,28,41,47);(1,3)(2,7)(4,12)(5,10)(6,14)(8,15)(9,16)(11,13)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45065);