Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18323 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3459 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3268 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);(3,4,8,13)(5,15,18,30)(6,12,7,21)(9,14,25,31)(10,22,28,16)(11,20,23,24)(17,33,19,26)(27,32,29,34);(3,5)(4,10)(6,16)(7,22)(8,18)(9,19)(11,26)(12,30)(13,28)(14,29)(15,21)(17,25)(20,32)(23,33)(24,34)(27,31);(3,4)(5,10)(6,11)(7,23)(8,13)(9,14)(12,24)(15,32)(16,26)(17,27)(18,28)(19,29)(20,21)(22,33)(25,31)(30,34);(3,6,8,7)(4,11,13,23)(5,16,18,22)(9,20,25,24)(10,26,28,33)(12,14,21,31)(15,27,30,29)(17,34,19,32)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254065);