#138756

order	512
nilpotency class	4
isoclinism	isoclinic class of #138694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

320 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=h,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=ehi,f^b=fh,h^b=hi,d^c=dhi,e^c=ehi,f^c=fhi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45952);