#508135

order	512
nilpotency class	4
isoclinism	isoclinic class of #508016
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	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

691 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi,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, 415331);