Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53277 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | 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)(5,7)(6,9)(8,10)(13,17)(16,18)(19,20,23,24)(21,25,27,31)(22,29,30,26)(28,34,33,32);(4,5)(8,9)(12,15)(16,18)(20,24)(21,27)(22,30)(32,34);(3,4,7,5)(6,8,10,9)(11,12)(13,16)(14,15)(17,18)(19,21)(20,25)(22,28)(23,27)(24,31)(26,32)(29,34)(30,33);(3,5,7,4)(6,9,10,8)(11,13)(12,16)(14,17)(15,18)(19,20)(21,25)(22,26)(23,24)(27,31)(28,32)(29,30)(33,34);(3,6,7,10)(4,8,5,9)(11,12)(13,16)(14,15)(17,18)(19,22,23,30)(20,26,24,29)(21,28,27,33)(25,32,31,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012878);