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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,10,12,7)(4,8,14,5)(11,13,16,15)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31)(33,34,38,42)(35,43,46,39)(36,41,37,40)(44,47,45,48);(1,3,6,12)(2,7,9,10)(4,11,14,16)(5,13,8,15)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,35,38,46)(34,39,42,43)(36,45,37,44)(40,48,41,47);(1,4)(2,8)(3,11)(5,9)(6,14)(7,15)(10,13)(12,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)(33,36,38,37)(34,40,42,41)(35,44,46,45)(39,47,43,48);(2,9)(4,14)(7,10)(11,16)(18,24)(19,27)(20,29)(30,31)(33,37,38,36)(34,41,42,40)(35,45,46,44)(39,48,43,47);(3,12)(7,10)(11,16)(13,15)(18,24)(19,27)(23,28)(26,32)(34,42)(39,43)(40,41)(47,48);(1,5,6,8)(2,4,9,14)(3,13,12,15)(7,11,10,16)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,d^2=i,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, 8012105);