Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10,15,19)(11,20,12,23)(13,18,14,17)(16,22,24,21);(2,4)(5,6)(7,8)(9,11,15,12)(10,16,19,24)(13,22,14,21)(17,23,18,20);(6,8)(9,12,15,11)(10,16,19,24)(13,21,14,22)(17,23,18,20)Polycyclic presentation:
2^7|a^2=g,b^2=g,c^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cf,d^b=dg,e^b=eg,d^c=dg,e^c=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7405, #7756, #7757, … |
| maximal normal subgroup of | #7405, #7756, #7757, … |
| maximal quotient of | #9957, #9959, #9967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 748);