Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
768 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(6,7)(9,17)(11,13)(15,20);(5,6,10,7)(8,13,16,11)(9,12,17,14)(15,19,20,18);(1,2)(3,4)(5,7,10,6)(8,13,16,11)(9,14,17,12)(15,19,20,18);(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19);(5,9)(6,12)(7,14)(8,15)(10,17)(11,18)(13,19)(16,20)Polycyclic presentation:
2^7|b^2=f,c^2=f,d^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| Frattini subgroup of | #94448 |
| Fitting subgroup of | #3487⋊C3, #11711509, #11713186, … |
| maximal subgroup of | #8464, #12708, #12714, … |
| maximal normal subgroup of | #8464, #12708, #12714, … |
| maximal quotient of | #34174, #34201, #34249, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2219);