Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29634 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#29711 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C2×#3459 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(4,5)(8,9)(11,12)(13,19)(14,17)(15,18)(16,21)(20,23)(22,25)(24,26);(3,4)(5,7)(6,8)(9,10)(11,13)(12,16)(14,20)(15,21)(17,24)(18,19)(22,26)(23,25);(3,5,7,4)(6,8,10,9)(11,14,15,22)(12,17,18,25)(13,20,21,26)(16,23,19,24);(3,6,7,10)(4,8,5,9)(14,22)(17,25)(20,26)(23,24)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342590);