Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 110 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 — 4 generators on 12 points:
(1,2)(3,4)(6,10)(7,8)(11,12);(5,6)(7,10)(8,12)(9,11);(2,4);(5,7,9,8)(6,10,11,12)Polycyclic presentation:
2^7|d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | #9347, #9365, #9368, … |
| maximal normal subgroup of | #9347, #9365, #9368, … |
| maximal quotient of | #18135, #18140, #18145, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1738);