#143846

order	512
nilpotency class	4
isoclinism	isoclinic class of #143821
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	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 335 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:

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

Polycyclic presentation:

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

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, 51042);