Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52330 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,9)(6,10)(7,8)(13,17)(16,18)(19,20,25,24)(21,27,22,26)(23,28,32,31)(29,34,30,33);(11,12)(13,16)(14,15)(17,18)(19,21,25,22)(20,26,24,27)(23,30,32,29)(28,34,31,33);(4,8)(5,10)(12,15)(16,18)(19,22,25,21)(20,27,24,26)(23,30,32,29)(28,34,31,33);(3,5)(4,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,18)(19,23)(20,28)(21,29)(22,30)(24,31)(25,32)(26,33)(27,34);(3,6)(4,5)(7,9)(8,10)(11,13)(12,16)(14,17)(15,18)(19,24,25,20)(21,27,22,26)(23,31,32,28)(29,34,30,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999575);