Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | Aut(G) = #72216771 (order 1024) · Inn(G) = C2×D16 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D16 |
| commutator, Frattini, cosocle | C16 | C23 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
56 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)(3,7)(4,16)(5,19)(6,8)(9,12)(10,13)(11,17)(14,26)(15,20)(18,32)(21,28)(22,27)(23,29)(24,30)(25,31);(2,7)(3,12)(4,13)(5,17)(8,20)(9,10)(11,26)(14,31)(15,29)(16,21)(18,30)(19,22)(23,24)(25,32)(27,28);(2,8)(7,20)(9,23)(10,24)(11,19)(16,32)(21,25)(22,26)Polycyclic presentation:
2^7|d^2=ef,e^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=de,e^a=ef,f^a=fg,d^b=de,e^b=ef,f^b=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | #13658, #13667, #13669, … |
| maximal normal subgroup of | #13658, #13667, #13669, … |
| maximal quotient of | #13616, #13617, #13634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 995);