Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) = #72231459 (order 2048) · Inn(G) = C2×D8 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | C23 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
75 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,14)(7,15,17,29)(8,20,22,35)(9,23,21,13)(11,24,25,18)(12,27,28,16)(19,34,33,31)(26,32,30,36);(5,7,16,32,9,19,22,24,10,17,27,36,21,33,8,18)(6,11,20,31,13,26,28,29,14,25,35,34,23,30,12,15);(2,4)(7,17)(11,25)(15,29)(18,24)(19,33)(26,30)(31,34)(32,36)Polycyclic presentation:
2^7|a^2=g,b^2=df,d^2=fg,f^2=g,b^a=bd,c^a=ce,d^a=df,f^a=fg,c^b=cg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7423, #7425, #13668, … |
| maximal normal subgroup of | #7423, #7425, #13668, … |
| maximal quotient of | #12434, #12443, #12455, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 951);