Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#51417 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6874 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)(7,8,12,15)(9,13,16,10)(11,14,19,21)(17,20,22,18)(24,29)(25,31)(27,35)(34,38);(9,16)(10,13)(11,19)(14,21)(23,24,28,29)(25,32,31,26)(27,36,35,30)(33,34,37,38);(3,4)(5,6)(7,9,12,16)(8,13,15,10)(11,17,19,22)(14,20,21,18)(23,25,28,31)(24,26,29,32)(27,33,35,37)(30,34,36,38);(7,10,12,13)(8,9,15,16)(11,18,19,20)(14,17,21,22)(23,26)(24,25)(27,34)(28,32)(29,31)(30,33)(35,38)(36,37);(7,11)(8,14)(9,17)(10,18)(12,19)(13,20)(15,21)(16,22)(23,27,28,35)(24,30,29,36)(25,33,31,37)(26,34,32,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614261);