Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) = C212 (order 4096) · Inn(G) = C24 · Out(G) = C28 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C24 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
228 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,10)(8,12)(9,11)(13,14,16,18)(15,19,20,17);(2,4)(5,7,9,12)(6,10,11,8)(13,14)(15,19)(16,18)(17,20);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11);(5,8)(6,7)(9,10)(11,12)(13,15,16,20)(14,17,18,19)Polycyclic presentation:
2^7|a^2=e,b^2=f,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de32 conjugacy classes — view the full character table.
| maximal subgroup of | #36708, #36713, #36728, … |
| maximal normal subgroup of | #36708, #36713, #36728, … |
| maximal quotient of | #14362, #14369, #14393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1422);