Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#45472 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6070 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)(5,7)(6,9)(8,10)(13,17)(16,18)(19,20,23,24)(21,27,29,25)(22,30,31,26)(28,32,34,33);(3,5,6,10)(4,7,9,8)(12,15)(13,17)(20,24)(22,31)(25,27)(28,34);(3,5)(4,8)(6,10)(7,9)(11,12)(13,16)(14,15)(17,18)(19,20)(21,25)(22,26)(23,24)(27,29)(28,32)(30,31)(33,34);(3,5)(4,8)(6,10)(7,9)(11,13)(12,16)(14,17)(15,18)(19,21,23,29)(20,25,24,27)(22,28,31,34)(26,32,30,33);(3,4)(5,8)(6,9)(7,10)(11,12)(13,16)(14,15)(17,18)(19,22,23,31)(20,26,24,30)(21,28,29,34)(25,32,27,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609420);