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) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C24 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
234 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(2,3)(5,6,8,10)(7,11,12,9)(15,19)(17,20);(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,16,18)(15,20,19,17);(5,6)(7,9)(8,10)(11,12)(13,15)(14,17)(16,19)(18,20);(1,3,4,2)(5,7,8,12)(6,9,10,11)Polycyclic presentation:
2^7|a^2=fg,b^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 | #27030, #27034, #27039, … |
| maximal normal subgroup of | #27030, #27034, #27039, … |
| maximal quotient of | #13876, #13921, #13974, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1228);