Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#482 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
234 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 100 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 — 4 generators on 18 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(12,15)(13,16)(17,18);(4,7)(8,10)(11,12,14,15)(13,16,17,18);(3,5,6,9)(4,8,7,10)(11,13,14,17)(12,16,15,18)38 conjugacy classes — view the full character table.
| maximal subgroup of | #9242, C4×#482, #21582, … |
| maximal normal subgroup of | #9242, C4×#482, #21582, … |
| maximal quotient of | C2×#1825, C2×#1854, C2×#1875, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1819);