Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(2,4)(6,9)(7,11)(10,12);(5,6,7,9,8,10,11,12);(1,2,3,4);(6,10)(9,12)Polycyclic presentation:
2^7|b^2=e,c^2=f,e^2=g,b^a=be,c^a=cf,e^a=eg,d^b=dg32 conjugacy classes — view the full character table.
| derived subgroup of | #72231265, #72240975, #72245425, … |
| Frattini subgroup of | #94788, #94791, #94792, … |
| maximal subgroup of | #9423, #9428, #9437, … |
| maximal normal subgroup of | #9423, #9428, #9437, … |
| maximal quotient of | #19621, #19625, #19629, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1883);