Finite Group DatabasebetaFinite Group Databasebeta| order | 270 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C4×C9 (order 72) · Inn(G) = C1 · Out(G) = C2×C4×C9 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C270 |
| abelian | C2×C5×C27 |
| multiplicative | (ℤ/271)ˣ, (ℤ/542)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C9 | C2×C3×C5 |
| socle, prime radical | C2×C3×C5 | C9 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 270 |
A minimal faithful permutation representation — 1 generator on 34 points:
(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29)(30,31,32,33,34)270 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(270, 4);