Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29610 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#29708 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2978 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)(13,17)(14,19)(15,23)(16,20)(18,21)(22,26)(24,25);(3,4)(5,7)(6,8)(9,10)(11,13,16,22)(12,17,20,26)(14,21,15,24)(18,23,25,19);(3,5,7,4)(6,8,10,9)(11,13)(12,18)(14,21)(15,24)(16,22)(17,23)(19,26)(20,25);(3,6,7,10)(4,8,5,9)(11,14,16,15)(12,19,20,23)(13,21,22,24)(17,18,26,25)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342603);