Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24808 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#24809 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,7,12,8,13,21,11)(5,14,17,9,19,15,18,25)(6,20,22,10,23,26,34,27)(16,30,31,28,33,24,32,29);(3,5,7,17,8,19,21,18)(4,9,11,14,13,25,12,15)(6,16,22,31,23,33,34,32)(10,24,20,28,27,30,26,29);(3,4)(5,15)(6,10)(7,11)(8,13)(9,17)(12,21)(14,19)(16,29)(18,25)(20,22)(23,27)(24,31)(26,34)(28,33)(30,32);(3,6,21,34,8,23,7,22)(4,10,12,26,13,27,11,20)(5,16,18,32,19,33,17,31)(9,24,15,29,25,30,14,28)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350254);