Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#546 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C24 |
| commutator, Frattini, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 14 points:
(1,2);(4,6)(7,8,11,12)(9,13,14,10);(8,12)(10,13);(3,4)(5,6)(7,9)(8,10)(11,14)(12,13);(7,10)(8,9)(11,13)(12,14)50 conjugacy classes — view the full character table.
| maximal subgroup of | #8359, #8367, #8381, … |
| maximal normal subgroup of | #8359, #8367, #8381, … |
| maximal quotient of | C2×D4×C22⋊C4, C2×#2339, C2×#2342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2195);