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×#46150 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6546 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)(7,8,11,14)(9,15,17,12)(10,18,20,13)(16,21,19,22)(23,24)(25,27)(26,30)(28,36)(29,32)(31,37)(33,38)(34,35);(7,9,11,17)(8,12,14,15)(10,19,20,16)(13,22,18,21)(23,25,29,26)(24,30,32,27)(28,33,37,34)(31,38,36,35);(3,4,5,6)(7,9)(8,12)(10,16)(11,17)(13,21)(14,15)(18,22)(19,20)(23,26,29,25)(24,30,32,27)(28,34,37,33)(31,38,36,35);(7,8)(9,12)(10,13)(11,14)(15,17)(16,21)(18,20)(19,22)(23,27,29,30)(24,25,32,26)(28,35,37,38)(31,33,36,34);(7,10,11,20)(8,13,14,18)(9,16,17,19)(12,21,15,22)(23,28,29,37)(24,31,32,36)(25,33,26,34)(27,35,30,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610025);