Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C24 |
| commutator | C22 | C23×C4 |
| prime radical | C24 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,7,8)(10,14)(11,17)(12,20)(22,24);(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22);(1,2)(3,4)(5,6)(7,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)Polycyclic presentation:
2^7|a^2=g,b^2=e,c^2=e,b^a=be,c^a=cf,d^a=de,c^b=ce44 conjugacy classes — view the full character table.
| Frattini subgroup of | #94171, #94175, #94488, … |
| maximal subgroup of | #8753, #8754, #9462, … |
| maximal normal subgroup of | #8753, #8754, #9462, … |
| maximal quotient of | #13776, #13786, #13795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1042);