Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#60344 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C24×C4 |
| prime radical | C2×#537 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6076 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)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20,25,27,24,28,22,32)(21,29,30,34,31,26,33,23);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21,24,31)(20,26,28,29)(22,33,25,30)(23,32,34,27);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(19,22,24,25)(20,27,28,32)(21,30,31,33)(23,26,34,29);(4,10)(8,11)(9,18)(16,17)(20,28)(21,31)(27,32)(30,33);(19,23,25,29,24,34,22,26)(20,21,27,30,28,31,32,33)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481749);