#144068

order	512
nilpotency class	4
isoclinism	isoclinic class of #143949
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#459
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

361 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 361 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18)(19,28)(20,34)(21,36)(22,41)(23,27)(24,30)(25,38)(26,37)(29,42)(31,47)(32,43)(33,48)(35,46)(39,44)(40,45);(17,19,23,31)(18,24,27,43)(20,35,21,39)(22,42,37,33)(25,29,45,48)(26,46,41,44)(28,36,47,34)(30,40,32,38);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,18)(19,29)(20,36)(21,34)(22,41)(23,27)(24,39)(25,40)(26,37)(28,33)(30,46)(31,48)(32,44)(35,43)(38,45)(42,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgi,g^a=gi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=dgi,e^c=eh,f^c=fg,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51264);