Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#20077 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3284 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4,7,10)(5,8,6,9)(11,12,17,15)(13,20,22,26)(14,19,16,23)(18,25,24,21);(3,5,7,6)(4,8,10,9)(11,13)(12,18)(14,21)(15,24)(16,25)(17,22)(19,26)(20,23);(4,9)(5,6)(8,10)(11,14,17,16)(12,19,15,23)(13,21,22,25)(18,26,24,20);(11,15,17,12)(13,20,22,26)(14,23,16,19)(18,25,24,21)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6266600);