Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | Aut(G) = D4×AGL(1,ℤ/16) (order 1024) · Inn(G) = #466 · Out(G) = C22×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 | #41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,8,14)(3,15,21,43)(4,22,25,39)(5,26,28,37)(6,13,7,12)(9,30,34,18)(10,29,36,24)(11,27,38,16)(17,46,49,63)(19,42,20,41)(23,54,55,35)(31,58,59,64)(32,50,33,44)(40,51,62,48)(45,53,56,61)(47,52,57,60);(1,3,16,44,7,20,28,30,8,21,27,50,6,19,5,18)(2,9,26,42,13,33,38,43,14,34,37,41,12,32,11,15)(4,17,45,64,24,48,55,57,25,49,56,58,29,51,23,47)(10,31,53,46,22,52,54,62,36,59,61,63,39,60,35,40);(1,4)(2,10)(3,17)(5,23)(6,29)(7,24)(8,25)(9,31)(11,35)(12,39)(13,22)(14,36)(15,40)(16,45)(18,47)(19,51)(20,48)(21,49)(26,53)(27,56)(28,55)(30,57)(32,60)(33,52)(34,59)(37,61)(38,54)(41,63)(42,46)(43,62)(44,64)(50,58)Polycyclic presentation:
2^7|a^2=g,b^2=df,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 | #13671, #13672, #13701, … |
| maximal normal subgroup of | #13671, #13672, #13701, … |
| maximal quotient of | #12431, #12434, #12440, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 954);