#143547

order	512
nilpotency class	4
isoclinism	isoclinic class of #143518
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	C₂×#493	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

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

normal subgroups only

This group has 365 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) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

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, 50743);