Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | Aut(G) = #72216771 (order 1024) · Inn(G) = #357 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #357 |
| commutator | C16 | C2×C4 |
| Frattini, cosocle | C2×C16 | C22 |
| prime radical | C22 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
35 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,8,13,7,31)(3,14,16,45,19,47,18,54)(4,20,23,55,26,57,25,62)(6,27,22,43,30,38,29,12)(9,32,33,15,35,49,41,51)(10,36,39,61,40,60,56,24)(17,52,48,37,53,64,28,46)(21,42,58,63,50,44,59,34);(1,3,8,19)(2,9,13,35)(4,21,26,50)(5,18,7,16)(6,28,30,48)(10,37,40,46)(11,41,31,33)(12,42,43,44)(14,20,47,57)(15,24,51,61)(17,22,53,29)(23,59,25,58)(27,34,38,63)(32,36,49,60)(39,52,56,64)(45,62,54,55)Polycyclic presentation:
2^7|a^2=d,b^2=g,c^2=efg,d^2=g,e^2=fg,f^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,c^b=ceg,d^b=dg,e^b=ef,f^b=fg26 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1420, #13625, #13657, … |
| maximal normal subgroup of | C2×#1420, #13625, #13657, … |
| maximal quotient of | #7386, #7421, #7445, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 152);