Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) = C22×#72232847 (order 8192) · Inn(G) = C23 · Out(G) = C23×#2846 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C23 |
| commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C24 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
264 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 116 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)(8,9)(11,12)(13,14,16,18)(15,19,20,17);(1,2)(3,4)(6,9)(10,12);(2,4)(13,15,16,20)(14,17,18,19);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^7|a^2=f,c^2=f,d^2=e,b^a=be,c^a=cf,c^b=cg44 conjugacy classes — view the full character table.
| maximal subgroup of | #9033, #9034, #11715, … |
| maximal normal subgroup of | #9033, #9034, #11715, … |
| maximal quotient of | #13844, #13846, #13848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1128);