Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) = C2×#72237160 (order 2048) · Inn(G) = C2×D8 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D8 |
| commutator | C8 | C22×C4 |
| Frattini, cosocle | C2×C8 | C23 |
| prime radical | C23 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(6,10)(7,11)(8,14)(12,16)(13,20)(15,19)(17,18);(5,6,11,18,8,13,15,16,9,12,19,20,14,17,7,10);(6,12)(10,16)(13,17)(18,20)Polycyclic presentation:
2^7|a^2=e,b^2=df,d^2=fg,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 | #12677, #12923, #12925, … |
| maximal normal subgroup of | #12677, #12923, #12925, … |
| maximal quotient of | #12009, #12025, #12035, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 891);