#6484613

order	512
nilpotency class	3
isoclinism	isoclinic class of #6473167
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)(20,34,38,62)(21,26,39,48)(22,41,35,27)(25,45,47,37)(30,53,56,60)(31,51,58,43)(33,44,54,52)(36,61,64,79)(40,66,63,49)(42,57,50,69)(46,70,71,65)(55,74,76,78)(59,68,75,73)(67,77,72,80);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19,22,33,23,32,35,54)(18,24,27,44,28,29,41,52)(20,30,37,57,38,56,45,69)(21,31,40,59,39,58,63,75)(25,42,34,60,47,50,62,53)(26,43,49,68,48,51,66,73)(36,55,65,77,64,76,70,80)(46,67,61,78,71,72,79,74);(3,9)(6,13)(10,11)(14,15)(17,20,35,45,23,38,22,37)(18,25,41,62,28,47,27,34)(19,30,54,69,32,56,33,57)(21,36,63,70,39,64,40,65)(24,42,52,53,29,50,44,60)(26,46,66,79,48,71,49,61)(31,55,75,80,58,76,59,77)(43,67,73,74,51,72,68,78);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,21,23,39)(18,26,28,48)(19,31,32,58)(20,36,38,64)(22,40,35,63)(24,43,29,51)(25,46,47,71)(27,49,41,66)(30,55,56,76)(33,59,54,75)(34,61,62,79)(37,65,45,70)(42,67,50,72)(44,68,52,73)(53,74,60,78)(57,77,69,80);(2,8)(6,13)(7,16)(14,15)(21,39)(26,48)(31,58)(36,64)(40,63)(43,51)(46,71)(49,66)(55,76)(59,75)(61,79)(65,70)(67,72)(68,73)(74,78)(77,80)

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,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, 6391809);