Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23 | C2×H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 80 points:
(1,2,6,3)(4,7,11,9)(5,10,13,8)(12,15,16,14)(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);(1,3,6,2)(4,7,11,9)(5,8,13,10)(12,15,16,14)(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);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(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);(4,11)(7,9)(12,16)(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);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(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=fhi,b^2=h,c^2=ghi,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=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7105847);