Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) = C212 (order 4096) · Inn(G) = C24 · Out(G) = C28 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
241 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 241 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,15)(11,12,16,14)(17,18,20,22)(19,23,24,21);(2,3)(6,7)(9,11,13,16)(10,14,15,12)(17,18)(19,21)(20,22)(23,24);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,19,20,24)(18,21,22,23);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^7|a^2=fg,b^2=ef,c^2=f,b^a=be,c^a=cf,d^a=de,c^b=cg38 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2510, #34754, #34788, … |
| maximal normal subgroup of | C2×#2510, #34754, #34788, … |
| maximal quotient of | #13883, #13901, #13999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1242);