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 | #52249 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4083 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)(8,15)(12,14)(13,16)(17,18,21,23)(19,25,26,22)(20,28,29,24)(27,30,32,31)(33,34,37,39)(35,38,41,46)(36,40,44,47)(42,45,43,48);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,18)(19,22)(20,28)(21,23)(24,29)(25,26)(27,31)(30,32)(33,35)(34,38)(36,43)(37,41)(39,46)(40,48)(42,44)(45,47);(1,4,7,9)(2,5,10,6)(3,12,15,16)(8,13,11,14)(17,19,21,26)(18,22,23,25)(20,27,29,32)(24,30,28,31)(34,39)(35,41)(36,44)(45,48);(1,5,7,6)(2,9,10,4)(3,13,15,14)(8,16,11,12)(18,23)(19,26)(24,28)(27,32)(33,36,37,44)(34,40,39,47)(35,42,41,43)(38,45,46,48);(1,5,7,6)(2,4,10,9)(3,14,15,13)(8,16,11,12)(17,20,21,29)(18,24,23,28)(19,27,26,32)(22,30,25,31)(34,39)(35,41)(40,47)(42,43);(1,6,7,5)(2,9,10,4)(3,14,15,13)(8,16,11,12)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=gh,e^2=g,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, 8012816);