Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C24 |
| commutator | C22 | C23×C4 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
280 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)(5,6,9,8)(7,11,12,10)(13,14,18,17)(15,20,16,19);(6,8)(10,11)(13,15,18,16)(14,19,17,20);(5,7)(6,10)(8,11)(9,12)(13,16,18,15)(14,19,17,20);(5,8,9,6)(7,11,12,10)(13,17,18,14)(15,19,16,20)Polycyclic presentation:
2^7|a^2=g,b^2=ef,c^2=ef,d^2=e,b^a=be,c^a=cf,d^b=de50 conjugacy classes — view the full character table.
| maximal subgroup of | #8770, #8771, #8778, … |
| maximal normal subgroup of | #8770, #8771, #8778, … |
| maximal quotient of | #13804, #13807, #13812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1089);