Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C25×#120 (order 1024) · Inn(G) = C23 · Out(G) = C27 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C23 |
| commutator, socle | C22 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| prime radical | #121 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
190 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,16,13,15,19,22)(11,14,17,21,18,20,23,24);(2,5)(6,8);(9,11,12,17,13,18,19,23)(10,14,16,21,15,20,22,24);(1,3,4,7)(2,6,5,8)(10,15)(14,20)(16,22)(21,24)Polycyclic presentation:
2^7|a^2=f,c^2=f,d^2=e,f^2=g,b^a=be,d^a=dg56 conjugacy classes — view the full character table.
| maximal subgroup of | #8247, #8248, #8249, … |
| maximal normal subgroup of | #8247, #8248, #8249, … |
| maximal quotient of | #20474, #20476, #20484, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1698);