#143361

order	512
nilpotency class	4
isoclinism	isoclinic class of #143209
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	C₂³	#449
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

365 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 80 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)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18)(19,29)(20,22)(21,36)(23,27)(24,28)(25,31)(26,39)(30,47)(32,42)(33,45)(34,46)(35,37)(38,44)(40,43)(41,48);(4,13)(7,12)(9,16)(11,14)(17,19,24,34)(18,25,28,41)(20,35,27,30)(21,37,40,47)(22,33,23,32)(26,42,44,45)(29,43,46,36)(31,38,48,39);(1,3)(2,6)(4,14)(5,10)(7,16)(8,15)(9,12)(11,13)(17,20,23,28,24,27,22,18)(19,30,33,41,34,35,32,25)(21,36,39,44,40,43,38,26)(29,31,42,37,46,48,45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50557);