Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#42891 |
| 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 | |
6770 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)(9,10,13,18)(11,14,20,22)(16,17,19,21)(23,24,29,28)(25,32,26,30)(27,35,36,31)(33,37,34,38);(7,9,12,13)(8,10,15,18)(11,19,20,16)(14,21,22,17)(23,25,29,26)(24,30,28,32)(27,33,36,34)(31,37,35,38);(3,4,5,6)(7,9)(8,10)(11,16)(12,13)(14,17)(15,18)(19,20)(21,22)(23,26,29,25)(24,30,28,32)(27,33,36,34)(31,38,35,37);(7,10)(8,13)(9,15)(11,17)(12,18)(14,19)(16,22)(20,21)(23,27,29,36)(24,31,28,35)(25,33,26,34)(30,37,32,38);(7,11,12,20)(8,14,15,22)(9,16,13,19)(10,17,18,21)(23,28,29,24)(25,30,26,32)(27,35,36,31)(33,37,34,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548732);