Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C24 |
| commutator | C22 | C23×C4 |
| prime radical | C23×D4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(5,6,7,8)(10,13)(11,15);(9,10,12,13)(11,14,15,16);(1,2)(3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16);(9,11)(10,14)(12,15)(13,16)Polycyclic presentation:
2^7|a^2=g,b^2=e,b^a=be,c^a=cf,d^a=de,c^b=ce44 conjugacy classes — view the full character table.
| derived subgroup of | #72236935, #72236936, #72241319, … |
| Frattini subgroup of | #94169, #94173, #94484, … |
| maximal subgroup of | #8748, #8749, #9497, … |
| maximal normal subgroup of | #8748, #8749, #9497, … |
| maximal quotient of | #13775, #13785, #13795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1041);