Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
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 | |
231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,6)(3,4,7,8)(10,13)(14,15)(17,18)(19,22)(20,23)(21,24);(2,6)(3,7)(9,10,12,13)(11,14,16,15)(17,19)(18,21)(20,24)(22,23);(1,3)(2,4)(5,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,22)(20,24)(21,23);(1,4)(2,7)(3,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,21)(20,23)(22,24)Polycyclic presentation:
2^7|a^2=fg,b^2=f,b^a=be,c^a=cf,c^b=cg,d^c=de32 conjugacy classes — view the full character table.
| Fitting subgroup of | #2643⋊C3 |
| maximal subgroup of | C2×#2643, #34570, #34584, … |
| maximal normal subgroup of | C2×#2643, #34570, #34584, … |
| maximal quotient of | #14259, #14265, #14331, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1375);