Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2,3,4)(5,6,7,8);(2,4)(10,12);(5,6)(7,8)(9,10)(11,12);(1,2)(3,4)(5,6)(7,8)Polycyclic presentation:
2^7|a^2=ef,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de32 conjugacy classes — view the full character table.
| derived subgroup of | #2679⋊C3, #7734⋊C3, #7822⋊C3, … |
| Fitting subgroup of | #2679⋊C3, #11714247, #11714255, … |
| maximal subgroup of | #7734, #7750, #7786, … |
| maximal normal subgroup of | #7734, #7750, #7786, … |
| maximal quotient of | #14357, #14440, #14535, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1411);