Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 32768 |
| 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 | C24 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,7,8)(10,14)(15,17)(16,22)(23,24);(2,4)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,23)(21,22);(1,2)(3,4)(5,6)(7,8)(9,11,13,19)(10,15,14,17)(12,20,21,18)(16,24,22,23);(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)Polycyclic presentation:
2^7|a^2=f,c^2=e,d^2=e,b^a=be,c^a=cf,c^b=cg,d^c=de32 conjugacy classes — view the full character table.
| maximal subgroup of | #9195, #9196, #11694, … |
| maximal normal subgroup of | #9195, #9196, #11694, … |
| maximal quotient of | #14258, #14261, #14274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1353);