Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(6,7)(9,10,13,16)(11,15,12,14);(9,11,13,12)(10,14,16,15);(5,6,8,7);(1,2,3,4)(5,7,8,6)(9,12,13,11)(10,15,16,14)Polycyclic presentation:
2^7|a^2=e,b^2=e,c^2=f,d^2=g,b^a=be,c^a=cf,d^a=dg44 conjugacy classes — view the full character table.
| Fitting subgroup of | #2868⋊C3, #11711259, #11711261, … |
| maximal subgroup of | #9691, #9692, #9694, … |
| maximal normal subgroup of | #9691, #9692, #9694, … |
| maximal quotient of | #14692, #14712, #14714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1600);