Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C24 |
| commutator, socle | C22 | C23×C4 |
| prime radical | #532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,13,10,12,18,19)(7,11,15,8,14,16,20,17);(2,4)(5,7)(6,11)(8,13)(9,15)(10,14)(12,16)(17,19)(18,20);(5,8,9,16,10,17,18,11)(6,7,13,15,12,14,19,20);(6,12)(7,14)(13,19)(15,20)Polycyclic presentation:
2^7|a^2=f,c^2=f,f^2=g,b^a=be,c^a=cg,d^a=dg,d^b=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #8878, #8880, #9047, … |
| maximal normal subgroup of | #8878, #8880, #9047, … |
| maximal quotient of | #17975, #17976, #17977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1664);