Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
244 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15);(2,5)(6,8)(10,14)(15,16);(10,15)(11,13)(14,16);(1,3,4,7)(2,6,5,8)(9,11,12,13)(10,16,14,15)Polycyclic presentation:
2^7|d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^c=dg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #9163, #9325, #9329, … |
| maximal normal subgroup of | #9163, #9325, #9329, … |
| maximal quotient of | #20738, #21122, #21365, … |
| automorphism group of | #360, #365 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1841);