Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) = #72232496 (order 2048) · Inn(G) = C22⋊C4 · Out(G) = C2×D42 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#328 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22⋊C4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C24 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
168 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 3 generators on 18 points:
(1,2);(3,4)(5,9,10,7)(6,8)(12,15,16,17);(3,5,6,10)(4,7,8,9)(11,12)(13,17)(14,16)(15,18)38 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#328, #8217, #8219, … |
| maximal normal subgroup of | C4×#328, #8217, #8219, … |
| maximal quotient of | C2×#1271, C2×#1272, #7566, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 231);