Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C23 |
| commutator | C22 | C2×C42 |
| socle, prime radical | C23 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3,4)(5,6,8,7);(2,4)(9,10,11,14,12,13,15,16);(5,7,8,6)(10,13)(14,16)Polycyclic presentation:
2^7|a^2=e,b^2=f,c^2=de,f^2=g,b^a=bd,c^b=cg56 conjugacy classes — view the full character table.
| maximal subgroup of | #7022, #7027, #7041, … |
| maximal normal subgroup of | #7022, #7027, #7041, … |
| maximal quotient of | #8261, #8263, #8267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 499);