#143651

order	512
nilpotency class	4
isoclinism	isoclinic class of #143626
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	#3169	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

345 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=gi,e^2=gh,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fhi,g^c=gi,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, 50847);