Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#27833 |
| 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 | |
1158 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);(3,4,9,14,10,16,8,15)(5,17,19,27,20,13,7,11)(6,21,25,30,24,29,23,12)(18,32,34,28,33,26,22,31);(3,5,9,19,10,20,8,7)(4,11,15,13,16,27,14,17)(6,22,23,33,24,34,25,18)(12,28,21,26,30,31,29,32);(3,6)(4,12)(5,18)(7,22)(8,23)(9,25)(10,24)(11,26)(13,28)(14,21)(15,29)(16,30)(17,31)(19,34)(20,33)(27,32);(3,7,8,20,10,19,9,5)(4,13,14,11,16,17,15,27)(6,22,23,33,24,34,25,18)(12,28,21,26,30,31,29,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365674);