Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) = C2×#511819 (order 1024) · Inn(G) = #124 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| socle | C2 | #394 |
| prime radical | #575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,10)(4,12)(5,13)(6,9)(7,11)(8,15)(14,16);(1,3)(2,7)(4,8)(5,14)(6,12)(9,16)(10,13)(11,15);(1,4,5,7,6,10,15,16)(2,3,8,11,9,12,13,14)Polycyclic presentation:
2^7|c^2=d,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fg,c^b=cf,d^b=dg,e^b=eg,e^c=eg,f^c=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | #7414, #7415, #12173, … |
| maximal normal subgroup of | #7414, #7415, #12173, … |
| maximal quotient of | #10456, #10457, #10458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 782);