#489570

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	#62325	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

674 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)(3,11)(4,10)(5,8)(6,9)(7,13)(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);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,16)(10,11)(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);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(17,20)(18,27)(19,34)(21,45)(22,50)(23,51)(24,43)(25,44)(26,53)(28,62)(29,64)(30,65)(31,40)(32,60)(33,66)(35,73)(36,74)(37,75)(38,71)(39,72)(41,47)(42,46)(48,58)(49,69)(52,70)(54,68)(55,76)(56,79)(57,80)(59,63)(61,78)(67,77);(2,9)(3,12)(8,10)(13,14)(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,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, 396766);