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 | |
4095 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)(6,9)(7,10)(8,14)(12,15)(13,16)(17,18)(19,21)(20,28)(22,26)(23,25)(24,30)(27,29)(31,32)(33,34,37,40)(35,41,43,38)(36,44,46,39)(42,47,45,48);(1,3)(2,8)(4,12)(5,13)(6,15)(7,14)(9,16)(10,11)(17,19)(18,23)(20,27)(21,26)(22,25)(24,29)(28,31)(30,32)(33,35)(34,38)(36,45)(37,43)(39,48)(40,41)(42,46)(44,47);(1,4,7,9)(2,5,10,6)(3,12,14,16)(8,13,11,15)(17,20,22,30)(18,24,26,28)(19,27,25,32)(21,31,23,29)(33,36,37,46)(34,39,40,44)(35,42,43,45)(38,47,41,48);(1,5,7,6)(2,9,10,4)(3,13,14,15)(8,16,11,12)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,28,32)(34,40)(35,43)(39,44)(42,45);(3,14)(8,11)(12,16)(13,15)(18,26)(19,25)(24,28)(27,32)(34,40)(38,41)(39,44)(47,48);(1,6,7,5)(2,9,10,4)(3,15,14,13)(8,16,11,12)Polycyclic presentation:
2^9|a^2=i,c^2=g,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, 8011644);