Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | #392 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
260 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)(9,10,13,11)(12,15,16,14)(17,18)(19,23)(20,21)(22,24);(2,5)(6,8)(10,11)(12,16)(18,21)(19,24);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,22)(20,24)(21,23);(1,3)(2,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^7|a^2=fg,c^2=ef,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de32 conjugacy classes — view the full character table.
| Fitting subgroup of | #2760⋊C3, #11711400, #11711441 |
| maximal subgroup of | #12187, #12263, #36776, … |
| maximal normal subgroup of | #12187, #12263, #36776, … |
| maximal quotient of | #14375, #14508, #14572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1492);