Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#29902 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,10)(6,12)(7,13)(8,11)(9,14)(15,25)(16,23)(17,24)(18,22)(19,28)(20,26)(21,27)(29,34)(30,32)(31,33);(3,5,8,15,7,18,9,17)(4,10,14,24,13,22,11,25)(6,16,20,29,19,30,21,31)(12,23,27,33,28,32,26,34);(4,11)(5,15)(6,19)(8,9)(10,22)(12,27)(13,14)(16,29)(17,18)(23,32)(26,28)(30,31);(3,6)(4,12)(5,16)(7,19)(8,20)(9,21)(10,23)(11,26)(13,28)(14,27)(15,29)(17,31)(18,30)(22,32)(24,33)(25,34)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340364);