Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C24 |
| commutator | C2 | C22×C42 |
| prime radical | C25 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(5,6,8,7)(9,10)(11,15)(12,13)(14,16);(1,2,3,4)(5,7,8,6)(10,13)(14,15);(9,10)(11,14)(12,13)(15,16);(9,11)(10,14)(12,16)(13,15)Polycyclic presentation:
2^7|a^2=f,b^2=g,b^a=be,d^a=de,c^b=ce68 conjugacy classes — view the full character table.
| derived subgroup of | #2273⋊C3, #10935⋊C3, #10936⋊C3, … |
| Fitting subgroup of | #2273⋊C3, #11714242, #11714243, … |
| maximal subgroup of | #8734, #9407, #9409, … |
| maximal normal subgroup of | #8734, #9407, #9409, … |
| maximal quotient of | #13749, #13775, #13778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1005);