Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) = C24×D42 (order 1024) · Inn(G) = #120 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #120 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C2×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
185 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,4)(5,6,8,10,9,7,12,11);(1,2,3,4)(5,7)(6,9)(8,10)(11,12);(5,6)(7,9)(8,10)(11,12)Polycyclic presentation:
2^7|a^2=f,b^2=dfg,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,e^a=eg,d^b=dg,f^b=fg,e^c=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1883, #20942, #20943, … |
| maximal normal subgroup of | C2×#1883, #20942, #20943, … |
| maximal quotient of | #8954, #8956, #8969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 615);