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) = C24 |
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 | |
252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,11)(8,9)(10,12)(13,14,17,19)(15,20,16,18);(2,4)(6,9)(7,12)(13,15,17,16)(14,18,19,20);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,16,17,15)(14,18,19,20);(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,18)(17,19)Polycyclic presentation:
2^7|a^2=ef,b^2=ef,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.
| maximal subgroup of | #7736, #7751, #7770, … |
| maximal normal subgroup of | #7736, #7751, #7770, … |
| maximal quotient of | #14384, #14440, #14446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1456);