Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(10,14)(11,17)(16,22)(19,20);(5,6,7,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,23)(21,22);(1,2)(3,4)(5,6)(7,8)(9,11)(10,15)(12,20)(13,17)(14,18)(16,24)(19,21)(22,23);(9,12,13,21)(10,16,14,22)(11,19,17,20)(15,23,18,24)Polycyclic presentation:
2^7|a^2=ef,b^2=eg,d^2=e,b^a=be,c^a=cf,c^b=cg,d^c=de32 conjugacy classes — view the full character table.
| Fitting subgroup of | #2658⋊C3, #11711255, #11711257, … |
| maximal subgroup of | #10094, #10102, #18437, … |
| maximal normal subgroup of | #10094, #10102, #18437, … |
| maximal quotient of | #14273, #14274, #14322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1390);