Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61901 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7572 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)(5,7)(6,8)(9,10)(11,12)(13,21)(14,18)(15,19)(16,20)(17,23)(22,26)(24,25);(4,7)(8,9)(11,13,16,23)(12,17,20,21)(14,22,15,24)(18,25,19,26);(4,8)(6,10)(7,9)(11,14,16,15)(12,18,20,19)(13,22,23,24)(17,25,21,26);(11,15,16,14)(12,19,20,18)(13,22,23,24)(17,25,21,26);(4,7)(8,9)(12,20)(17,21)(18,19)(25,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485775);