Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C24 |
| commutator | C2 | C24×C4 |
| Frattini, socle, cosocle | C22 | C25 |
| prime radical | C22×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19);(7,14)(10,13)(15,20)(17,19);(6,11)(10,13)(12,18)(17,19);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)Polycyclic presentation:
2^7|a^2=g,e^2=f,b^a=bf,d^a=df,c^b=cf68 conjugacy classes — view the full character table.
| Fitting subgroup of | #3428⋊C3, #11711311, #11711315, … |
| maximal subgroup of | #10743, #10747, #10752, … |
| maximal normal subgroup of | #10743, #10747, #10752, … |
| maximal quotient of | #33996, #33999, C4×#518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2160);