Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) = #72231699 (order 1024) · Inn(G) = C2×D8 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D8 |
| commutator | C8 | C22×C4 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C2 | C2×#102 |
| prime radical | C2×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,7,11,6,21)(3,12,15,29,13,28,22,26)(4,16,18,27,20,31,19,9)(8,17,24,30,23,32,25,14);(1,3)(2,8)(4,17)(5,15)(6,22)(7,13)(9,26)(10,24)(11,23)(12,16)(14,19)(18,30)(20,32)(21,25)(27,29)(28,31);(3,13)(8,23)(12,28)(14,30)(15,22)(17,32)(24,25)(26,29)Polycyclic presentation:
2^7|a^2=e,d^2=fg,e^2=g,f^2=g,b^a=bd,d^a=df,f^a=fg,c^b=cg,d^b=df,f^b=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #12649, #12651, #12740, … |
| maximal normal subgroup of | #12649, #12651, #12740, … |
| maximal quotient of | #11592, #11593, #11594, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 875);