Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60600 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,17,15)(13,16,20,18)(14,21,24,26)(19,25,22,23);(4,7)(8,10)(11,13)(12,18)(14,21)(15,16)(17,20)(19,25)(22,23)(24,26);(11,14,17,24)(12,19,15,22)(13,21,20,26)(16,23,18,25);(3,5,6,9)(4,8,7,10)(11,15,17,12)(13,16,20,18)(14,22,24,19)(21,23,26,25);(11,16,17,18)(12,20,15,13)(14,23,24,25)(19,26,22,21)122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482331);