#7186731

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021375
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#532	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1572 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:

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

Polycyclic presentation:

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

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, 7093927);