#151063

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	C₂×#366
prime radical	C₂³	#366
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

129 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 129 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 80 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,30)(20,26)(21,39)(22,43)(23,45)(24,29)(25,33)(27,31)(28,40)(32,52)(34,65)(35,67)(36,55)(37,68)(38,51)(41,71)(42,70)(44,72)(46,60)(47,74)(48,64)(49,57)(50,59)(53,56)(54,62)(58,69)(61,77)(63,73)(66,80)(75,78)(76,79);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,31,56,22,34,37,60,20,32,57,78,38,61,42,64,24,36,59,79,44,66,41,63,23,35,58,74,40,62,21,33)(18,25,39,54,28,47,69,67,45,73,71,80,72,76,50,55,29,48,70,77,51,75,49,52,26,46,68,65,43,53,27,30);(3,9)(6,12)(10,15)(13,16)(17,20,24,23)(18,26,29,45)(19,32,36,35)(21,37,42,41)(22,38,44,40)(25,46,48,73)(27,49,50,69)(28,43,51,72)(30,52,55,67)(31,57,59,58)(33,60,64,63)(34,61,66,62)(39,68,70,71)(47,53,75,76)(54,65,77,80)(56,78,79,74)

Polycyclic presentation:

2^9|a^2=e,b^2=df,c^2=i,d^2=fh,e^2=g,f^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cg

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58259);