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×#18331 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3460 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3230 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);(3,4,9,15)(5,17,19,30)(6,13,7,23)(8,14,10,16)(11,24,28,18)(12,22,25,21)(20,27,26,34)(29,33,31,32);(3,5)(4,11)(6,18)(7,24)(8,26)(9,19)(10,20)(12,27)(13,30)(14,31)(15,28)(16,29)(17,23)(21,33)(22,32)(25,34);(3,4,10,16)(5,11,20,29)(6,12,22,23)(7,25,21,13)(8,14,9,15)(17,18,27,32)(19,28,26,31)(24,34,33,30);(3,6,9,7)(4,12,15,25)(5,18,19,24)(8,21,10,22)(11,27,28,34)(13,16,23,14)(17,31,30,29)(20,32,26,33)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254074);