Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 8192 |
| 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 | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
274 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 112 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:
(5,6,9,7)(8,11,12,10)(14,17)(15,19);(2,4)(6,7)(10,11)(13,14,16,17)(15,18,19,20);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,17)(19,20);(5,8,9,12)(6,10,7,11)(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^7|a^2=ef,b^2=f,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | #26370, #26375, #26379, … |
| maximal normal subgroup of | #26370, #26375, #26379, … |
| maximal quotient of | #13885, #13920, #13979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1186);