Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | Aut(G) = C2×#3109 (order 256) · Inn(G) = #442 · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #442 |
| commutator | C8 | C22×C4 |
| Frattini, cosocle | C2×C8 | C23 |
| prime radical | #135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
71 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)(3,13,15,24)(4,17)(5,9)(6,21)(7,12)(8,14,25,31)(10,18)(11,23)(16,28,22,30)(19,29,32,26)(20,27);(1,3)(2,8)(4,14)(5,19)(6,22)(7,15)(9,24)(10,28)(11,29)(12,25)(13,27)(16,23)(17,30)(18,31)(20,32)(21,26);(1,4,6,5,7,18,23,20)(2,9,11,17,12,27,21,10)(3,14,22,19,15,31,16,32)(8,24,29,30,25,13,26,28)Polycyclic presentation:
2^7|a^2=e,c^2=ef,d^2=f,f^2=g,b^a=bd,c^a=cf,d^a=df,f^a=fg,d^b=dfg,e^b=eg,f^b=fg,d^c=dg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2180, #33468, #33469, … |
| maximal normal subgroup of | C2×#2180, #33468, #33469, … |
| maximal quotient of | #12065, #12066, #12077, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 912);