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×#50898 |
| 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 | |
6750 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)(9,13,16,21)(10,11,14,20)(17,22,19,18)(23,24)(25,27)(26,30)(28,36)(29,32)(31,37)(33,38)(34,35);(9,16)(10,14)(11,20)(13,21)(23,25,29,26)(24,30,32,27)(28,33,37,34)(31,38,36,35);(3,4,5,6)(7,9)(8,13)(10,19)(11,18)(12,16)(14,17)(15,21)(20,22)(23,26,29,25)(24,30,32,27)(28,34,37,33)(31,38,36,35);(7,10,12,14)(8,11,15,20)(9,17,16,19)(13,22,21,18)(23,27,29,30)(24,25,32,26)(28,35,37,38)(31,33,36,34);(7,11)(8,14)(9,18)(10,15)(12,20)(13,17)(16,22)(19,21)(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, 7613923);