Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60767 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
8272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,15,20)(13,21,24,26)(14,19,16,18)(17,25,22,23);(3,5)(4,7)(6,10)(8,9)(11,12)(13,22)(14,18)(15,20)(16,19)(17,24)(21,25)(23,26);(5,10)(7,9)(11,13)(12,17)(14,23)(15,24)(16,25)(18,21)(19,26)(20,22);(4,8)(7,9)(11,14,15,16)(12,18,20,19)(13,23,24,25)(17,21,22,26);(11,14,15,16)(12,19,20,18)(13,23,24,25)(17,26,22,21)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482628);