Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) = C22×#72232847 (order 8192) · Inn(G) = C24 · Out(G) = C24×#120 |
| 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 | |
292 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 292 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:
(2,4)(5,6)(7,9)(8,11)(10,12)(14,17)(15,19);(6,7)(10,11)(13,14,16,17)(15,18,19,20);(1,2)(3,4)(5,7,9,6)(8,11,12,10);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^7|b^2=eg,c^2=g,b^a=be,c^a=cf,d^a=de,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | #9517, #9518, #11577, … |
| maximal normal subgroup of | #9517, #9518, #11577, … |
| maximal quotient of | #13923, #13960, #14015, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1311);