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) = C24 |
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 | |
255 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,9)(10,12)(13,14,17,18)(15,16,19,20);(1,2,3,4)(6,9)(10,12)(14,18)(15,19);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,19)(18,20);(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,20)(18,19)Polycyclic presentation:
2^7|a^2=e,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×#2558, #34829, #34866, … |
| maximal normal subgroup of | C2×#2558, #34829, #34866, … |
| maximal quotient of | #13917, #13949, #14008, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1290);