#137879

order	512
nilpotency class	3
isoclinism	isoclinic class of #9159
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	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂×#41	M₁₆
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 303 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45075);