Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 16384 |
| 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 | C24 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
218 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)(10,11)(14,15)(17,18)(19,22)(20,23)(21,24);(2,6)(3,7)(9,10,13,11)(12,14,16,15)(17,19)(18,21)(20,24)(22,23);(1,3)(2,4)(5,7)(6,8)(9,11,13,10)(12,14,16,15)(17,19)(18,22)(20,24)(21,23);(1,4)(2,7)(3,6)(5,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,21)(20,23)(22,24)Polycyclic presentation:
2^7|a^2=g,b^2=f,c^2=f,b^a=be,c^a=cf,c^b=cg,d^c=de32 conjugacy classes — view the full character table.
| maximal subgroup of | #10344, #10345, #10346, … |
| maximal normal subgroup of | #10344, #10345, #10346, … |
| maximal quotient of | #14265, #14275, #14329, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1378);