Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | Aut(G) = D4×#442 (order 512) · Inn(G) = #466 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| prime radical | #472 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,8)(3,14)(4,17)(5,6)(9,18)(10,25)(11,22)(12,21)(13,23)(15,30)(16,19)(20,26)(27,32)(28,31);(1,2,9,21,6,12,4,8,7,13,17,22,5,11,18,23)(3,10,24,32,14,26,15,20,16,27,29,25,19,28,30,31);(1,3)(2,10)(4,15)(5,19)(6,14)(7,16)(8,20)(9,24)(11,28)(12,26)(13,27)(17,29)(18,30)(21,32)(22,25)(23,31)Polycyclic presentation:
2^7|b^2=dfg,d^2=fg,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fg,e^c=eg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2223, #33617, #33618, … |
| maximal normal subgroup of | C2×#2223, #33617, #33618, … |
| maximal quotient of | #12432, #12435, #12441, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 955);