#489642

order	512
nilpotency class	4
isoclinism	isoclinic class of #489373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#466	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=ghi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396838);