#143860

order	512
nilpotency class	4
isoclinism	isoclinic class of #143811
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

369 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,9)(4,12)(5,8)(6,10)(7,14)(11,16)(13,15)(19,28)(20,33)(21,22)(24,39)(25,41)(26,38)(29,44)(30,35)(31,45)(32,46)(34,47)(36,37)(40,48)(42,43);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,18)(19,29)(20,34)(21,38)(22,26)(23,27)(24,30)(25,37)(28,44)(31,40)(32,43)(33,47)(35,39)(36,41)(42,46)(45,48);(4,14)(6,15)(7,12)(10,13)(17,19,23,32)(18,24,27,40)(20,35,37,45)(21,31,22,30)(25,42,34,44)(26,43,38,29)(28,36,46,33)(39,47,48,41)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51056);