Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) = #23794 (order 256) · Inn(G) = C42 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C42 |
| commutator | C4 | C4×C8 |
| Frattini, cosocle | C2×C16 | C22 |
| socle | C2 | #346 |
| prime radical | C22 | M32 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,8,4,10,13,23,5,11,14,21,15,25,27,31,6,9,12,22,16,24,26,32,17,7,18,19,28,20,29,30);(2,7,9,11)(3,12)(8,21,22,19)(10,20,24,25)(13,26)(14,18)(23,31,32,30)(27,29)Polycyclic presentation:
2^7|a^2=c,b^2=d,c^2=e,e^2=f,f^2=g,b^a=bf,d^a=dg,c^b=cg44 conjugacy classes — view the full character table.
| maximal subgroup of | #7333, #7389, #7390, … |
| maximal normal subgroup of | #7333, #7389, #7390, … |
| maximal quotient of | #7330, #7332, #7466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 130);