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) = C22×C4 |
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 | |
189 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 189 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18,20,22)(19,23,24,21);(2,5)(3,8)(9,11,13,16)(10,14,12,15)(17,18)(19,23)(20,22)(21,24);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,14,16,15);(1,3,4,8)(2,6,5,7)(9,11)(10,15)(12,14)(13,16)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^7|a^2=eg,b^2=f,c^2=f,d^2=ef,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 | #36759, #36764, #36767, … |
| maximal normal subgroup of | #36759, #36764, #36767, … |
| maximal quotient of | #14413, #14414, #14425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1525);