#138534

order	512
nilpotency class	4
isoclinism	isoclinic class of #138488
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

335 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,6,10,14)(4,11,12,7)(5,8)(9,13)(15,16)(17,18,23,30,22,46,44,28,24,31,45,60,25,29,21,43)(19,32,36,26,49,52,34,51,37,62,69,53,35,59,48,54)(20,38,41,55,47,57,73,71,42,58,72,75,40,27,39,56)(33,63,66,70,65,61,64,50,67,76,80,78,68,79,74,77);(1,3,5,10)(2,6)(4,9,12,15)(7,13)(8,14)(11,16)(17,19,25,49,24,37,22,35)(18,26,46,54,31,53,29,51)(20,33,40,65,42,67,47,68)(21,36,45,34,44,69,23,48)(27,50,38,70,57,77,58,78)(28,59,60,62,43,52,30,32)(39,66,72,64,73,80,41,74)(55,63,71,79,75,76,56,61);(1,4,5,12)(2,7,8,11)(3,9,10,15)(6,13,14,16)(17,20,24,42)(18,27,31,57)(19,33,37,67)(21,39,44,73)(22,47,25,40)(23,41,45,72)(26,50,53,77)(28,55,43,75)(29,58,46,38)(30,56,60,71)(32,61,62,79)(34,64,48,74)(35,68,49,65)(36,66,69,80)(51,78,54,70)(52,76,59,63)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=g,d^2=ghi,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egi,f^b=fh,h^b=hi,d^c=dgi,e^c=ei,f^c=fgi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45730);