#143934

order	512
nilpotency class	4
isoclinism	isoclinic class of #143911
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	#469
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

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

normal subgroups only

This group has 323 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)(3,10)(4,15)(5,8)(6,9)(7,12)(11,14)(13,16)(17,18,23,29)(19,31,20,34)(21,30,24,27)(22,38,40,46)(25,35,43,32)(26,33,45,36)(28,37,42,41)(39,47,48,44);(1,3,6,14)(2,7,9,16)(4,13,5,12)(8,11,15,10)(17,19,23,20)(18,25,29,43)(21,35,24,32)(22,36,40,33)(26,28,45,42)(27,34,30,31)(37,48,41,39)(38,47,46,44);(1,2)(3,11)(4,8)(5,15)(6,9)(7,12)(10,14)(13,16)(17,20,23,19)(18,26,29,45)(21,35,24,32)(22,39,40,48)(25,42,43,28)(27,44,30,47)(31,38,34,46)(33,41,36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51130);