Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C23 |
| commutator | C22 | C2×C42 |
| socle | C23 | C22×C4 |
| prime radical | C24 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24);(2,4)(9,11,12,17,13,18,19,23)(10,14,16,21,15,20,22,24);(5,7,8,6)(10,15)(14,20)(16,22)(21,24)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=e,f^2=g,b^a=bd,c^a=cg56 conjugacy classes — view the full character table.
| maximal subgroup of | #7214, #7216, #8348, … |
| maximal normal subgroup of | #7214, #7216, #8348, … |
| maximal quotient of | #8148, #8149, #8154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 482);