Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 12288 |
| 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 | |
761 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)(5,6,10,11)(7,13,15,8)(9,17,18,12)(14,16,20,19);(6,11)(8,13)(12,17)(16,19);(1,2)(3,4)(5,7)(6,8)(9,14)(10,15)(11,13)(12,16)(17,19)(18,20);(5,8)(6,7)(9,16)(10,13)(11,15)(12,14)(17,20)(18,19);(5,9,10,18)(6,12,11,17)(7,14,15,20)(8,16,13,19)Polycyclic presentation:
2^7|a^2=f,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| Fitting subgroup of | #3489⋊C3, #11711058, #11713182, … |
| maximal subgroup of | #8467, #17606, #17608, … |
| maximal normal subgroup of | #8467, #17606, #17608, … |
| maximal quotient of | #34175, #34202, #34203, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2221);