Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#577 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C25 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
730 conjugacy classes of subgroups; only the 428 normal subgroups are shown.
This group has 730 conjugacy classes of subgroups — too many to draw. Only its 428 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) = 32 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,7,11)(5,9,12,16)(6,8,14,10)(13,15,18,17);(4,8)(6,14)(9,15)(10,11)(13,18)(16,17);(3,5)(4,9)(6,13)(7,12)(8,15)(10,17)(11,16)(14,18);(5,12)(9,16)(13,18)(15,17)44 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#577⋊C3, #11712960, #11712967, … |
| maximal subgroup of | #22456, #22459, #22464, … |
| maximal normal subgroup of | #22456, #22459, #22464, … |
| maximal quotient of | C2×#2897, C2×#2898, C2×#2912, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2314);