#136771

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 366 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) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43967);