Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#44278 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2456 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5544 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);(5,9)(7,10)(13,17)(16,18)(19,20)(21,25)(22,26)(23,24);(3,4)(5,10)(6,8)(7,9)(11,12,15,14)(13,16,17,18)(20,24)(22,25);(3,4,6,8)(5,7,9,10)(11,13,15,17)(12,16,14,18)(19,21,23,26)(20,25,24,22);(3,4)(5,7)(6,8)(9,10)(19,22,23,25)(20,26,24,21);(3,5)(4,7)(6,9)(8,10)(11,14,15,12)(13,18,17,16)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7575464);