Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,16)(12,21)(13,23)(14,18)(15,19)(17,22)(20,24);(1,2,3,4)(6,8)(9,11,13,19,14,20,22,12)(10,15,17,16,18,21,23,24);(9,12,22,20,14,19,13,11)(10,16,23,15,18,24,17,21)Polycyclic presentation:
2^7|a^2=f,b^2=df,c^2=eg,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7113, #13014, #13015, … |
| maximal normal subgroup of | #7113, #13014, #13015, … |
| maximal quotient of | #9656, #9662, #9718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 701);