Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#59974 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6770 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)(5,9)(6,8)(7,10)(12,17)(15,18)(20,24)(21,26);(3,4,6,8)(5,7,10,9)(11,12,16,17)(13,18,14,15)(20,24)(21,26);(11,13,16,14)(12,18,17,15)(19,20)(21,25)(22,26)(23,24);(3,4)(5,7)(6,8)(9,10)(11,14,16,13)(12,15,17,18)(19,21,23,26)(20,25,24,22);(3,5)(4,7)(6,10)(8,9)(11,15)(12,13)(14,17)(16,18)(19,22,23,25)(20,26,24,21)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573186);