Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | Aut(G) = #152710 (order 512) · Inn(G) = #354 · Out(G) = D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #354 |
| commutator | C42 | C2×C4 |
| Frattini, cosocle | C4⋊Q8 | C22 |
| prime radical | C22 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ G | |
| LECS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
39 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,6,11,18,28)(3,12,14,30,13,29,31,26)(5,10,20,27,21,8,17,25)(7,22,23,15,24,16,32,19);(1,3,6,13)(2,7)(4,16,18,22)(5,19,21,15)(8,12)(9,26)(10,29)(11,24)(14,17,31,20)(23,25)(27,32)(28,30)Polycyclic presentation:
2^7|a^2=d,b^2=f,c^2=fg,d^2=g,e^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,c^b=cf,d^b=def,e^b=eg,d^c=dfg,e^d=eg14 conjugacy classes — view the full character table.
| maximal subgroup of | #7525, #7526, C2×#1413, … |
| maximal normal subgroup of | #7525, #7526, C2×#1413, … |
| maximal quotient of | #7358, #7367, #7413, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 145);