Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C23 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,23,22,19);(1,3,4,8)(2,5,6,7)(10,14)(15,22)(16,21)(17,24);(9,11,13,19,12,18,20,23)(10,15,16,17,14,22,21,24);(11,18)(15,22)(17,24)(19,23)Polycyclic presentation:
2^7|a^2=e,b^2=eg,c^2=f,f^2=g,b^a=be,c^a=cf,f^a=fg,d^c=dg32 conjugacy classes — view the full character table.
| maximal subgroup of | #9440, #10023, #10025, … |
| maximal normal subgroup of | #9440, #10023, #10025, … |
| maximal quotient of | #20801, #21159, #21323, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1898);