Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#27834 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | #2495 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1148 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);(3,4,10,14,9,16,8,15)(5,17,26,28,19,13,7,11)(6,20,24,12,25,31,23,32)(18,27,21,29,33,34,22,30);(3,5,8,7,9,19,10,26)(4,11,14,17,16,28,15,13)(6,21,24,33,25,22,23,18)(12,29,20,27,32,30,31,34);(3,6)(4,12)(5,18)(7,22)(8,23)(9,25)(10,24)(11,27)(13,30)(14,31)(15,20)(16,32)(17,29)(19,33)(21,26)(28,34);(3,7,8,19,9,26,10,5)(4,13,14,11,16,17,15,28)(6,22,23,33,25,21,24,18)(12,30,31,27,32,29,20,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365675);