Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C24 |
| commutator | C2 | C22×C42 |
| socle | C22 | C23×C4 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,12,9,10,15,17)(7,11,14,18,13,16,19,20);(1,2,3,4)(6,10)(11,16)(12,17)(18,20);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20);(7,13)(11,16)(14,19)(18,20)Polycyclic presentation:
2^7|a^2=e,b^2=f,e^2=g,b^a=bg,d^c=dg68 conjugacy classes — view the full character table.
| Fitting subgroup of | #2875⋊C3, #11711202, #11711203, … |
| maximal subgroup of | #9311, #9312, #12609, … |
| maximal normal subgroup of | #9311, #9312, #12609, … |
| maximal quotient of | #17316, #17320, #17329, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1607);