Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
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 | |
236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(7,10)(8,12)(13,14,18,17)(15,20,19,16);(5,6,9,11)(7,8,10,12)(13,15)(14,16)(17,20)(18,19);(1,2,3,4)(5,7,9,10)(6,8,11,12)(13,16,18,20)(14,19,17,15);(5,8)(6,10)(7,11)(9,12)(13,17,18,14)(15,20,19,16)Polycyclic presentation:
2^7|a^2=e,b^2=g,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=dg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7660, #7663, #7680, … |
| maximal normal subgroup of | #7660, #7663, #7680, … |
| maximal quotient of | #14687, #14699, #14707, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1591);