Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) = C22×#1202 (order 4096) · Inn(G) = C25 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
571 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,8)(4,7)(5,6)(10,13)(14,16);(2,6)(7,8);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,13)(15,16);(1,2)(3,7)(4,8)(5,6);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^7|c^2=f,e^2=g,b^a=bf,c^a=cg,e^a=ef,d^b=df44 conjugacy classes — view the full character table.
| maximal subgroup of | #8412, #17652, #17679, … |
| maximal normal subgroup of | #8412, #17652, #17679, … |
| maximal quotient of | #34191, #34229, #34232, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2246);