Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | #525 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
369 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 369 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,13)(8,16)(9,11)(10,14)(12,18)(15,20)(17,19);(5,7,9,15)(6,10,11,19)(8,17,18,14)(12,20,16,13);(2,4);(6,11)(10,19)(12,16)(13,20)Polycyclic presentation:
2^7|b^2=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | #9519, #18136, #18146, … |
| maximal normal subgroup of | #9519, #18136, #18146, … |
| maximal quotient of | #18133, #18138, #18143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1736);