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×#29903 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2336 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)(5,14)(6,19)(7,24)(8,12)(9,13)(10,16)(11,21)(15,32)(17,26)(18,31)(20,25)(22,28)(23,29)(27,33)(30,34);(3,5,7,17,9,16,8,18)(4,10,12,26,13,14,24,31)(6,20,21,15,23,34,22,33)(11,27,19,30,28,32,29,25);(3,6,9,23)(4,11,13,28)(5,15,16,33)(7,22,8,21)(10,25,14,30)(12,29,24,19)(17,20,18,34)(26,32,31,27);(3,4)(5,10)(6,11)(7,12)(8,24)(9,13)(14,16)(15,30)(17,26)(18,31)(19,21)(20,27)(22,29)(23,28)(25,33)(32,34)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340365);