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) = C22×C4 |
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 | |
234 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,7)(4,5)(6,8)(9,10,13,12)(11,15,16,14)(18,21)(19,23);(2,5)(6,7)(11,16)(14,15)(17,18,20,21)(19,22,23,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,15,16,14)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^7|a^2=eg,b^2=efg,d^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×#2576, #34849, #34861, … |
| maximal normal subgroup of | C2×#2576, #34849, #34861, … |
| maximal quotient of | #13917, #13947, #13984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1308);