#97313

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

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

normal subgroups only

This group has 453 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:

(1,2,6,11,7,10,5,15)(3,12,13,16,4,8,14,9)(17,18)(19,28)(20,32,34,41)(21,35)(22,26)(23,27)(24,30)(25,33,42,37)(29,44,47,38)(31,39)(36,40)(43,46,48,45)(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);(1,3,7,4)(2,8,10,12)(5,13,6,14)(9,11,16,15)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,38)(26,39)(27,40)(28,35)(32,43)(33,46)(34,47)(37,45)(41,48)(42,44)(49,51,54,62)(50,55,58,70)(52,60,65,77)(53,66,68,61)(56,69,63,79)(57,72,74,59)(64,78,67,76)(71,75,73,80);(1,4,7,3)(2,9,10,16)(5,13,6,14)(8,11,12,15)(17,20,23,34)(18,25,27,42)(19,29,31,47)(21,33,22,37)(24,38,40,44)(26,41,35,32)(28,43,39,48)(30,45,36,46)(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,b^2=gh,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, 4509);