Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46343 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5218 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)(8,10)(12,14)(15,18)(19,20,23,22)(21,25,26,24);(4,6)(5,9)(11,12,16,14)(13,18,17,15);(3,4,7,6)(5,8,9,10)(11,13)(12,15)(14,18)(16,17)(19,21,23,26)(20,24,22,25);(3,5,7,9)(4,8,6,10)(11,14,16,12)(13,18,17,15)(19,22,23,20)(21,25,26,24);(3,6,7,4)(5,10,9,8)(11,15,16,18)(12,17,14,13)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610213);