Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) = #72237540 (order 1024) · Inn(G) = C2×D4 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×D4 |
| commutator | C4 | C22×C8 |
| Frattini, cosocle | C2×C8 | C23 |
| socle | C2 | C2×C4⋊C8 |
| prime radical | C23 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,9,19,24,7,11,20,25,4,17,18,30)(3,12,15,27,16,22,29,31,13,26,28,32,14,8,21,23);(1,3,4,14,7,13,6,16)(2,8,9,12,11,22,17,26)(5,15,18,21,20,28,19,29)(10,23,24,27,25,31,30,32);(3,13)(8,22)(12,26)(14,16)(15,28)(21,29)(23,31)(27,32)Polycyclic presentation:
2^7|a^2=e,b^2=d,d^2=g,e^2=f,f^2=g,b^a=bd,d^a=dg,c^b=cg44 conjugacy classes — view the full character table.
| maximal subgroup of | #11989, #11990, #11991, … |
| maximal normal subgroup of | #11989, #11990, #11991, … |
| maximal quotient of | #11997, #12005, #12014, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 885);