Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) = #72231730 (order 2048) · Inn(G) = C2×D8 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D8 |
| commutator | C8 | C22×C4 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C2 | C2×#107 |
| prime radical | C23 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
70 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)(3,12,16,28,14,26,17,31)(4,18,19,25,20,32,13,9)(8,22,24,30,23,29,27,15);(1,3,13,29,6,17,20,30,7,14,19,22,5,16,4,15)(2,8,18,28,10,24,25,26,11,23,32,31,21,27,9,12);(3,14)(8,23)(12,26)(15,30)(16,17)(22,29)(24,27)(28,31)Polycyclic presentation:
2^7|a^2=e,b^2=df,d^2=fg,e^2=g,f^2=g,b^a=bd,d^a=df,f^a=fg,c^b=cg32 conjugacy classes — view the full character table.
| maximal subgroup of | #12678, #12924, #12927, … |
| maximal normal subgroup of | #12678, #12924, #12927, … |
| maximal quotient of | #12009, #12025, #12032, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 893);