Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| prime radical | #525 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(5,8)(9,10,14,17)(11,12,19,15)(13,21,22,16)(18,23,24,20);(1,2,4,6)(3,5,7,8)(9,11,14,19)(10,15,17,12)(13,18,22,24)(16,23,21,20);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,24)(22,23);(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,23)(22,24);(9,13,14,22)(10,16,17,21)(11,18,19,24)(12,20,15,23)Polycyclic presentation:
2^7|a^2=f,b^2=g,e^2=f,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| maximal subgroup of | #8468, #17605, #17609, … |
| maximal normal subgroup of | #8468, #17605, #17609, … |
| maximal quotient of | #34180, #34202, #34203, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2227);