#143854

order	512
nilpotency class	4
isoclinism	isoclinic class of #143803
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	#449
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

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

normal subgroups only

This group has 359 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)(3,9)(4,12)(5,8)(6,10)(7,14)(11,16)(13,15)(17,18,20,25)(19,28,30,47)(21,35,34,46)(22,27,23,39)(24,40,41,31)(26,37,42,38)(29,36,44,45)(32,43,33,48);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,18)(19,29)(20,25)(21,26)(22,39)(23,27)(24,32)(28,45)(30,44)(31,43)(33,41)(34,42)(35,38)(36,47)(37,46)(40,48);(4,14)(6,15)(7,12)(10,13)(17,19,20,30)(18,24,25,41)(21,36,34,45)(22,33,23,32)(26,43,42,48)(27,44,39,29)(28,46,47,35)(31,37,40,38)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,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,f^b=fi,d^c=dg,e^c=eh,f^c=fi,h^c=hi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51050);