Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | Aut(G) = #501097 (order 512) · Inn(G) = #353 · Out(G) = D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #353 |
| commutator | C42 | C2×C4 |
| Frattini, cosocle | #124 | C22 |
| prime radical | C2×D4 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ G | |
| LECS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
49 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)(3,11,13,26)(4,15,16,31)(5,18,20,9)(6,10)(7,12,21,28)(8,17)(14,24,19,27)(22,29,23,30)(25,32);(1,3,6,14)(2,7,10,23)(4,12,16,30)(5,19,20,13)(8,24,25,11)(9,21,18,22)(15,27,31,26)(17,28,32,29)Polycyclic presentation:
2^7|a^2=d,b^2=g,c^2=f,e^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,c^b=cf,d^b=defg,e^b=eg,d^c=dfg,e^d=eg14 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1407, #13577, #13578, … |
| maximal normal subgroup of | C2×#1407, #13577, #13578, … |
| maximal quotient of | #7353, #7372, #7403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 139);