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 | #58019 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4068 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)(3,11)(4,5)(6,10)(7,9)(8,14)(12,13)(15,16)(17,18)(19,25)(20,22)(21,28)(23,26)(24,30)(27,31)(29,32);(1,3)(2,8)(4,12)(5,15)(6,13)(7,14)(9,11)(10,16)(17,19)(18,23)(20,25)(21,29)(22,26)(24,32)(27,30)(28,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,4,7,10)(2,5,9,6)(3,12,14,16)(8,15,11,13)(17,20,22,18)(19,25,26,23)(21,28,30,24)(27,32,29,31)(33,35,37,43)(34,38,40,41)(36,44,45,42)(39,47,48,46);(2,9)(4,10)(8,11)(12,16)(18,20)(19,26)(24,28)(27,29)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(1,5,7,6)(2,4,9,10)(3,13,14,15)(8,16,11,12)(17,21,22,30)(18,24,20,28)(19,27,26,29)(23,31,25,32)(34,40)(38,41)(39,48)(46,47);(1,6,7,5)(2,10,9,4)(3,13,14,15)(8,16,11,12)Polycyclic presentation:
2^9|c^2=gi,d^2=i,e^2=g,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, 8012122);