#140317

order	512
nilpotency class	4
isoclinism	isoclinic class of #12483
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1925
prime radical	C₂³	#482
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

222 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=d,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,e^c=egh,h^c=hi

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, 47513);