Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#45767 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2813 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(4,6)(9,15)(13,18)(16,21)(20,22)(23,24,29,27)(25,32,26,30)(28,35,36,31)(33,37,34,38);(8,12)(9,15)(10,17)(20,22)(23,25,29,26)(24,30,27,32)(28,33,36,34)(31,37,35,38);(3,4)(5,6)(7,8,11,12)(9,13,15,18)(10,14,17,19)(16,20,21,22)(23,26,29,25)(24,30,27,32)(28,34,36,33)(31,37,35,38);(7,9)(8,13)(10,16)(11,15)(12,18)(14,20)(17,21)(19,22)(23,27,29,24)(25,30,26,32)(28,35,36,31)(33,37,34,38);(7,10)(8,14)(9,16)(11,17)(12,19)(13,20)(15,21)(18,22)(23,28)(24,31)(25,33)(26,34)(27,35)(29,36)(30,37)(32,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609624);