Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | Aut(G) order 1024 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | #121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
160 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,25)(24,28)(26,30)(27,29)(31,32);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,26)(18,22,25,30)(20,27,28,32)(23,29,24,31);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(2,8)(6,9)(7,16)(14,15)(19,26)(22,30)(27,32)(29,31)Polycyclic presentation:
2^7|b^2=eg,c^2=e,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,c^b=ce,d^b=dg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3243, #60859, #60867, … |
| maximal normal subgroup of | C2×#3243, #60859, #60867, … |
| maximal quotient of | #21843, #21871, #21991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1975);