#136801

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	C₂×C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 355 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)(3,11,14,15)(4,8,7,13)(5,9)(6,10)(12,16)(17,18,21,26,23,28,30,39,22,27,37,45,20,25,35,43)(19,29,33,46,32,42,47,44,34,40,36,48,31,24,38,41);(1,3,12,13,6,14,5,8)(2,7,16,11,10,4,9,15)(17,19,30,38,20,31,21,36,22,34,35,47,23,32,37,33)(18,24,39,48,25,40,26,44,27,42,43,46,28,29,45,41);(1,4,6,7)(2,8,10,13)(3,9,14,16)(5,11,12,15)(17,20,22,23)(18,25,27,28)(19,31,34,32)(21,35,37,30)(24,40,42,29)(26,43,45,39)(33,38,36,47)(41,48,44,46)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,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, 43997);