Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#343 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22⋊C4 |
| commutator | C22 | C2×C42 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C24 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2);(7,8,9,12,10,11,13,14);(3,4,5,6)(8,11)(9,13)44 conjugacy classes — view the full character table.
| maximal subgroup of | #7251, C4×#343, #10727, … |
| maximal normal subgroup of | #7251, C4×#343, #10727, … |
| maximal quotient of | C2×#1280, #7908, #7909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 473);