Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C24 |
| commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
547 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,13)(15,16);(1,3)(2,6)(4,7)(5,8);(1,2,5,4)(3,6,8,7)(10,13)(14,16);(3,8)(6,7);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^7|a^2=f,c^2=f,e^2=fg,b^a=bf,c^a=cg,d^b=df50 conjugacy classes — view the full character table.
| Fitting subgroup of | #3482⋊C3, #11711052, #11711808, … |
| maximal subgroup of | #8442, #17518, #17519, … |
| maximal normal subgroup of | #8442, #17518, #17519, … |
| maximal quotient of | #34127, #34128, #34129, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2214);