Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19496 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,6)(7,10)(8,9)(11,12)(13,22)(14,18)(15,20)(16,21)(17,25)(19,24)(23,26);(3,5,6,4,8,10,7,9)(11,13,16,14)(12,18,21,22)(15,24,17,23)(19,20,26,25);(11,14,16,13)(12,19,21,26)(15,24,17,23)(18,20,22,25);(3,6,8,7)(4,5,9,10)(11,15,16,17)(12,20,21,25)(13,23,14,24)(18,19,22,26)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261763);