Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
186 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(3,7)(9,10,14,12)(11,16,15,13)(19,23)(21,24);(1,2,5,4)(3,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,18,20,22)(19,24,23,21);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,16,15,13)(17,19,20,23)(18,21,22,24);(1,4,5,2)(3,8,7,6)(9,13,14,16)(10,15,12,11)Polycyclic presentation:
2^7|a^2=efg,b^2=f,c^2=ef,d^2=e,b^a=be,c^a=cf,d^a=de,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | #27063, #27067, #27070, … |
| maximal normal subgroup of | #27063, #27067, #27070, … |
| maximal quotient of | #13885, #13921, #13989, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1246);