Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51452 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,7,9)(5,6,10,8)(11,12,15,18)(13,16,14,17)(19,20,24,27)(21,25,28,22)(23,26,31,33)(29,32,34,30);(3,5,7,10)(4,8,9,6)(11,13,15,14)(12,16,18,17)(21,28)(22,25)(23,31)(26,33);(11,12)(13,16)(14,17)(15,18)(19,21,24,28)(20,25,27,22)(23,29,31,34)(26,32,33,30);(3,5)(4,6)(7,10)(8,9)(19,22,24,25)(20,21,27,28)(23,30,31,32)(26,29,33,34);(3,6)(4,5)(7,8)(9,10)(11,14,15,13)(12,17,18,16)(19,23)(20,26)(21,29)(22,30)(24,31)(25,32)(27,33)(28,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614210);