Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #56544 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,10)(4,14)(5,8)(6,9)(7,12)(11,15)(13,16)(17,18)(19,25)(20,28)(21,23)(22,27)(24,30)(26,31)(29,32)(33,34)(35,41)(36,44)(37,40)(38,43)(39,46)(42,47)(45,48);(1,3,6,12)(2,7,9,10)(4,11,5,13)(8,15,14,16)(17,19,21,27)(18,22,23,25)(20,29,30,26)(24,32,28,31)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,2)(3,7)(4,14)(5,8)(6,9)(10,12)(11,16)(13,15)(18,23)(20,30)(22,25)(26,29)(33,36,37,46)(34,39,40,44)(35,42,43,45)(38,47,41,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,21,30)(18,24,23,28)(19,26,27,29)(22,31,25,32)(34,40)(35,43)(39,44)(42,45);(3,12)(7,10)(11,13)(15,16)(18,23)(19,27)(24,28)(26,29)(34,40)(38,41)(39,44)(47,48);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012112);