#97310

order	512
nilpotency class	3
isoclinism	isoclinic class of #97300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

451 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,e^2=g,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=eg,f^c=fh

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, 4506);