Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) = D4×#442 (order 512) · Inn(G) = #447 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | #576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,13)(3,14,17,30)(4,18,22,29)(5,12,6,11)(8,24,27,15)(9,23,28,16)(10,20,25,21)(19,31,32,26);(1,3)(2,8)(4,19)(5,16)(6,23)(7,17)(9,25)(10,28)(11,26)(12,31)(13,27)(14,18)(15,21)(20,24)(22,32)(29,30);(1,3)(2,9)(4,15)(5,23)(6,16)(7,17)(8,25)(10,27)(11,30)(12,14)(13,28)(18,31)(19,21)(20,32)(22,24)(26,29)Polycyclic presentation:
2^7|a^2=g,d^2=fg,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fg,d^b=df,f^b=fg,e^c=eg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2194, #33538, #33539, … |
| maximal normal subgroup of | C2×#2194, #33538, #33539, … |
| maximal quotient of | #12111, #12113, #12167, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 926);