Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C2×#1202 (order 2048) · Inn(G) = C23 · Out(G) = C23×#120 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C23 |
| commutator, socle | C22 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×#41 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
206 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,11,13,14,16,15);(2,5)(6,8);(9,11,12,14,13,15,16,10);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^7|a^2=f,c^2=f,d^2=e,f^2=g,b^a=be,c^a=cg56 conjugacy classes — view the full character table.
| maximal subgroup of | #8194, #8195, #8196, … |
| maximal normal subgroup of | #8194, #8195, #8196, … |
| maximal quotient of | #20472, #20473, C4×#435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1697);