Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) = C22×#3109 (order 512) · Inn(G) = #102 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #102 |
| commutator | C8 | C2×C8 |
| Frattini, cosocle | C4×C8 | C22 |
| prime radical | C2×D8 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10,12,16,13,18,23,17)(11,19,21,15,22,14,20,24);(10,14)(11,20)(12,23)(15,16)(17,24)(18,19)(21,22)Polycyclic presentation:
2^7|a^2=d,c^2=e,d^2=f,e^2=g,b^a=bc,c^a=ce,e^a=eg,c^b=ceg,d^b=dg,e^b=eg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1333, #11597, #11598, … |
| maximal normal subgroup of | C2×#1333, #11597, #11598, … |
| maximal quotient of | #7070, #7074, #7112, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 65);