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) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| prime radical | #121 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,8)(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23);(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23);(1,2)(3,6)(4,5)(7,8);(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23);(1,3,4,7)(2,6,5,8)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)Polycyclic presentation:
2^7|a^2=f,b^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| Fitting subgroup of | #3523⋊C3, #11711039, #11712018, … |
| maximal subgroup of | #8417, #17549, #17601, … |
| maximal normal subgroup of | #8417, #17549, #17601, … |
| maximal quotient of | #34196, #34235, #34243, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2255);