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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
231 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 231 conjugacy classes of subgroups — too many to draw. Only its 94 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)(10,14)(11,12)(15,16);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,16)(13,15);(9,11,13,12)(10,14,15,16)Polycyclic presentation:
2^7|c^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=dg,f^a=fg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #9348, #9366, #9367, … |
| maximal normal subgroup of | #9348, #9366, #9367, … |
| maximal quotient of | #20702, #20846, #20912, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1826);