Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#539 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C24 |
| commutator, Frattini, cosocle | C22 | C25 |
| prime radical | C2×#121 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
572 conjugacy classes of subgroups; only the 396 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 396 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) = 64 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4)(5,6)(7,8)(9,10)(13,17)(15,18);(4,8)(5,9)(11,12,14,16)(13,15,17,18);(3,5,7,9)(4,6,8,10)(11,13)(12,15)(14,17)(16,18);(3,6)(4,9)(5,8)(7,10)44 conjugacy classes — view the full character table.
| maximal subgroup of | #8407, #17387, #17406, … |
| maximal normal subgroup of | #8407, #17387, #17406, … |
| maximal quotient of | C2×#2313, C2×#2318, C2×#2322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2183);