#6484622

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473174
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6391818);