Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C2×#72219291 (order 4096) · Inn(G) = C23 · Out(G) = C22×#2679 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#435 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C23 |
| commutator | C22 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| socle | C23 | C22×C4 |
| prime radical | C22×D4 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
282 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 14 points:
(1,2);(3,4)(5,6)(7,8,10,9,11,12,14,13);(4,6);(7,9,10,12,11,13,14,8)56 conjugacy classes — view the full character table.
| maximal subgroup of | #8161, #8610, #8613, … |
| maximal normal subgroup of | #8161, #8610, #8613, … |
| maximal quotient of | C2×#1581, C2×#1583, C2×#1582, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1659);