#143539

order	512
nilpotency class	4
isoclinism	isoclinic class of #143528
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	#3147	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

340 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50735);