Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 24576 |
| 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 | C2×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,15,16,13);(2,5)(6,7);(9,11,12,16)(10,13,14,15);(1,2)(3,6)(4,5)(7,8);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^7|a^2=g,c^2=g,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 | #3503⋊C3, #11711043, #11711044, … |
| maximal subgroup of | #8424, #8425, #8473, … |
| maximal normal subgroup of | #8424, #8425, #8473, … |
| maximal quotient of | #34183, #34214, #34216, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2235);