Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#410 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊C4 |
| commutator | C22 | C23×C4 |
| Frattini, cosocle | C23 | C24 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | C22×#41 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
394 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,11,13,8)(6,9)(7,10)(12,17,18,15)(14,16);(3,5)(4,8)(6,12)(7,13)(9,15)(10,11)(14,18)(16,17);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)44 conjugacy classes — view the full character table.
| maximal subgroup of | #10742, #10743, #10748, … |
| maximal normal subgroup of | #10742, #10743, #10748, … |
| maximal quotient of | C2×#1458, C2×#1460, C2×#1461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1614);