Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#532 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C23 |
| commutator, Frattini, cosocle | C22 | C25 |
| socle | C23 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
864 conjugacy classes of subgroups; only the 452 normal subgroups are shown.
This group has 864 conjugacy classes of subgroups — too many to draw. Only its 452 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 — 5 generators on 14 points:
(1,2);(4,6)(7,8)(9,12)(10,11)(13,14);(8,11)(12,14);(3,4,5,6);(7,9,10,13)(8,12,11,14)56 conjugacy classes — view the full character table.
| derived subgroup of | #72227997, #72228001, #72228012, … |
| Frattini subgroup of | #94427, #94429, #94431, … |
| maximal subgroup of | #8313, #8332, #8337, … |
| maximal normal subgroup of | #8313, #8332, #8337, … |
| maximal quotient of | C2×#2285, C2×#2293, C2×#2297, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2174);