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) = C23×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 | |
236 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(5,6,9,8)(7,11,12,10)(15,18)(16,20);(2,4)(6,8)(7,12)(13,14,17,19)(15,16,18,20);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(13,15,17,18)(14,16,19,20);(5,8,9,6)(7,11,12,10)(13,16)(14,18)(15,19)(17,20)Polycyclic presentation:
2^7|a^2=e,b^2=ef,c^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 | #36741, #36743, #36748, … |
| maximal normal subgroup of | #36741, #36743, #36748, … |
| maximal quotient of | #14386, #14398, #14403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1452);