Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #27987 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#27989 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3058 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1870 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);(3,4,8,14,9,15,23,13)(5,16,11,10,19,21,18,6)(7,22,24,12,25,28,34,29)(17,20,27,30,32,26,31,33);(3,5,8,11,9,19,23,18)(4,10,13,16,15,6,14,21)(7,17,24,27,25,32,34,31)(12,26,22,30,29,20,28,33);(3,6,9,10)(4,11)(5,14)(7,20,25,26)(8,21,23,16)(12,27)(13,19)(15,18)(17,28)(22,32)(24,33,34,30)(29,31);(3,7,23,34,9,25,8,24)(4,12,14,28,15,29,13,22)(5,17,18,31,19,32,11,27)(6,20,16,30,10,26,21,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365728);