Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#72220637 (order 3072) · Inn(G) = C22 · Out(G) = C2×#41×S4 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×#504 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C22 |
| commutator | C2 | C23×C8 |
| Frattini, cosocle | C8 | C24 |
| socle | C22 | C22×C8 |
| prime radical | C2×#41 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,6,10,7,11,16,22,8,12,17,23,18,24,28,32)(5,9,14,20,15,21,27,31,13,19,25,29,26,30,33,34);(3,5)(4,9)(6,14)(7,15)(8,13)(10,20)(11,21)(12,19)(16,27)(17,25)(18,26)(22,31)(23,29)(24,30)(28,33)(32,34);(5,13)(9,19)(14,25)(15,26)(20,29)(21,30)(27,33)(31,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2138);