Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | Aut(G) = #483875 (order 512) · Inn(G) = #367 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #367 |
| commutator | C16 | C2×C4 |
| Frattini, cosocle | C2×C16 | C22 |
| prime radical | C22 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(2,7,9,20)(3,12)(4,16)(5,17)(8,23)(10,24,27,32)(11,30,28,22)(13,18)(14,26)(15,25)(19,31,29,21);(1,2,8,24,4,10,3,7,17,28,12,21,13,19,23,30,6,9,26,32,18,27,15,20,5,11,25,31,16,29,14,22)Polycyclic presentation:
2^7|a^2=d,b^2=cef,c^2=efg,e^2=fg,f^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,d^b=dg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1426, #13638, #13657, … |
| maximal normal subgroup of | C2×#1426, #13638, #13657, … |
| maximal quotient of | #7463, #7474, C2×#1426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 158);