Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 16384 |
| 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 | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
263 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 263 conjugacy classes of subgroups — too many to draw. Only its 100 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:
(5,6,9,10)(7,8,11,12)(15,19)(17,20);(2,4)(6,10)(7,11)(13,14,16,18)(15,20,19,17);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,17)(16,19)(18,20);(5,8)(6,11)(7,10)(9,12)Polycyclic presentation:
2^7|a^2=e,b^2=f,b^a=be,c^a=cf,c^b=cg,d^c=de32 conjugacy classes — view the full character table.
| maximal subgroup of | #18232, #18233, #18234, … |
| maximal normal subgroup of | #18232, #18233, #18234, … |
| maximal quotient of | #14256, #14259, #14263, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1355);