Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#390 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C23 |
| commutator | C22 | C23×C4 |
| Frattini, cosocle | C23 | C24 |
| prime radical | C23×D4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
680 conjugacy classes of subgroups; only the 236 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 236 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);(4,6)(8,9)(11,12,14,13);(7,8,10,9)(11,13,14,12);(3,4,5,6)(7,9,10,8)56 conjugacy classes — view the full character table.
| derived subgroup of | #72338279, #72338282, #72339622, … |
| Frattini subgroup of | #94086, #94090, #94177, … |
| maximal subgroup of | C2×#1270, #7572, #7592, … |
| maximal normal subgroup of | C2×#1270, #7572, #7592, … |
| maximal quotient of | C2×C4⋊#95, C2×#1437, C4×#390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1024);