Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62032 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6672 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);(4,7)(5,9)(11,12,15,20)(13,22,24,17)(14,21,16,19)(18,25,26,23);(3,4)(5,8)(6,7)(9,10)(11,13)(12,17)(14,25)(15,24)(16,23)(18,21)(19,26)(20,22);(11,13)(12,18)(14,23)(15,24)(16,25)(17,21)(19,22)(20,26);(4,7)(8,10)(11,14,15,16)(12,19,20,21)(13,23,24,25)(17,18,22,26);(3,5,6,9)(4,8,7,10)(11,14,15,16)(12,19,20,21)(13,23,24,25)(17,18,22,26)98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10486022);