Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#504 (order 128) · Inn(G) = C22 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C1 |
| modular maximal cyclic | M128 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C32 | C22 |
| commutator, socle | C2 | C2×C32 |
| prime radical | C22 | C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
19 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64);(2,34)(4,36)(6,38)(8,40)(10,42)(12,44)(14,46)(16,48)(18,50)(20,52)(22,54)(24,56)(26,58)(28,60)(30,62)(32,64)Polycyclic presentation:
2^7|a^2=c,c^2=d,d^2=e,e^2=f,f^2=g,b^a=bg80 conjugacy classes — view the full character table.
| maximal subgroup of | #7511, #7513, #7514, … |
| maximal normal subgroup of | #7511, #7513, #7514, … |
| maximal quotient of | #7510, C22⋊C64, C4⋊C64, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 160);