Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#535 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C24 |
| commutator, Frattini, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
948 conjugacy classes of subgroups; only the 436 normal subgroups are shown.
This group has 948 conjugacy classes of subgroups — too many to draw. Only its 436 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) = 32 |
A minimal faithful permutation representation — 5 generators on 14 points:
(1,2);(4,6)(8,11)(9,13);(7,8,10,11)(9,12,13,14);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14);(7,9)(8,12)(10,13)(11,14)44 conjugacy classes — view the full character table.
| derived subgroup of | #72237051, #72237052, #72237053, … |
| Frattini subgroup of | #94420, #94423, #94424, … |
| maximal subgroup of | #8516, #8522, #10973, … |
| maximal normal subgroup of | #8516, #8522, #10973, … |
| maximal quotient of | C2×#2309, C2×#2311, C2×#2317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2178);