#144255

order	512
nilpotency class	4
isoclinism	isoclinic class of #144219
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	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 387 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) = 80
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51451);