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×#56333 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2714 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5184 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);(11,12,14,16)(13,17,18,15)(19,20,24,28)(21,22,26,25)(23,31,33,27)(29,34,32,30);(3,4)(5,9)(6,8)(7,10)(13,18)(15,17)(19,21,24,26)(20,25,28,22)(23,32,33,29)(27,30,31,34);(3,4,6,8)(5,7,10,9)(11,13)(12,15)(14,18)(16,17)(19,21)(20,22)(23,29)(24,26)(25,28)(27,30)(31,34)(32,33);(3,4)(5,7)(6,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,26)(21,28)(23,30)(24,25)(27,32)(29,31)(33,34);(3,5)(4,7)(6,10)(8,9)(19,23,24,33)(20,27,28,31)(21,29,26,32)(22,30,25,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006140);