Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#55864 |
| 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 | |
5054 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)(5,10,6,8)(11,12)(13,17)(14,16)(15,18)(19,20)(21,24)(22,27)(23,28)(25,29)(26,31)(30,32)(33,34);(3,5)(4,8)(6,7)(9,10)(11,13,16,15)(12,17,14,18)(20,21)(22,29)(26,28)(30,34);(5,6)(8,10)(11,14,16,12)(13,17,15,18)(19,21,24,20)(22,27,29,25)(23,28,31,26)(30,33,34,32);(3,6,7,5)(4,8,9,10)(11,15,16,13)(12,17,14,18)(19,22)(20,25)(21,27)(23,30)(24,29)(26,32)(28,33)(31,34);(3,4)(5,10)(6,8)(7,9)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009917);