Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10,12,13)(11,14,15,16)(17,18)(19,22)(20,21)(23,24);(2,5)(6,8)(10,13)(11,15)(18,20)(19,23);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,23)(18,22,20,24);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,24,23,22)Polycyclic presentation:
2^7|a^2=efg,c^2=fg,d^2=ef,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de32 conjugacy classes — view the full character table.
| maximal subgroup of | #7745, #7755, #7782, … |
| maximal normal subgroup of | #7745, #7755, #7782, … |
| maximal quotient of | #14440, #14457, #14471, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1536);