Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C24 |
| commutator | C22 | C23×C4 |
| prime radical | C2×#127 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(5,6,8,7)(9,10,13,11)(12,15,16,14);(5,7,8,6)(10,11)(14,15);(1,2,3,4)(9,11,13,10)(12,15,16,14);(9,12,13,16)(10,14,11,15)Polycyclic presentation:
2^7|a^2=g,b^2=eg,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=ce44 conjugacy classes — view the full character table.
| Frattini subgroup of | #72292975, #72292976, #72292978, … |
| maximal subgroup of | #9101, #9105, #9295, … |
| maximal normal subgroup of | #9101, #9105, #9295, … |
| maximal quotient of | #13779, #13785, #13791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1060);