#490328

order	512
nilpotency class	4
isoclinism	isoclinic class of #490099
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

649 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:

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

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=dgh,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397524);