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 | #52277 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,10,7)(5,8,13,12)(11,15,16,14)(17,18)(19,22)(20,25)(21,26)(23,28)(24,30)(27,31)(29,32)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,3,6,10)(2,7,4,9)(5,11,13,16)(8,14,12,15)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,35,37,43)(34,38,40,41)(36,44,45,42)(39,47,48,46);(1,4,6,2)(3,9,10,7)(5,12,13,8)(11,15,16,14)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(2,4)(7,9)(8,12)(14,15)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(34,40)(38,41)(39,48)(46,47);(3,10)(7,9)(11,16)(14,15)(18,19)(23,25)(24,26)(31,32);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=h,f^2=ghi,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, 8012810);