Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 1474560 |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | C2×#585 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C24 |
| commutator, Frattini, cosocle | C2 | C26 |
| socle | C22 | C25 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
2978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 18 points:
(1,2);(3,4)(5,11)(6,10)(7,9)(8,12)(13,17)(14,16)(15,18);(3,5)(4,8)(6,13)(7,12)(9,11)(10,15)(14,18)(16,17);(5,12)(8,11)(13,18)(15,17);(4,9)(8,11)(10,16)(15,17);(3,6,7,14)(4,10,9,16)(5,13,12,18)(8,15,11,17)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2325);