Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | Aut(G) = #72232676 (order 2048) · Inn(G) = C2×C22⋊C4 · Out(G) = C2×H(5,𝔽2) |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C2×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
163 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 3 generators on 16 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,13)(12,14)(15,16);(1,3)(2,5)(4,8)(6,7)(10,13)(11,16)(14,15);(10,14)(13,15)Polycyclic presentation:
2^7|a^2=e,d^2=g,b^a=bd,c^a=cg,d^a=df,d^b=dg,e^b=efg26 conjugacy classes — view the full character table.
| maximal subgroup of | #8487, #8516, #8517, … |
| maximal normal subgroup of | #8487, #8516, #8517, … |
| maximal quotient of | #7592, #7593, #7599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 242);