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 |
| prime radical | #508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
52 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)(3,13)(4,18)(5,24)(6,28)(7,9)(8,14)(10,21)(11,20)(12,26)(15,45)(16,48)(17,40)(19,37)(22,56)(23,39)(25,35)(27,58)(29,41)(30,44)(31,52)(32,51)(33,42)(34,46)(36,63)(38,43)(47,64)(49,54)(50,55)(53,62)(57,60)(59,61);(1,3,7,17)(2,8,9,30)(4,19,23,53)(5,25,27,59)(6,29,26,16)(10,35,33,61)(11,37,34,62)(12,38,28,32)(13,39,40,18)(14,42,44,21)(15,46,47,20)(22,57,55,52)(24,31,58,60)(36,64,49,45)(41,50,48,56)(43,54,51,63);(2,9)(8,30)(10,33)(11,34)(12,28)(13,40)(18,39)(24,58)(31,60)(32,38)(35,61)(36,49)(37,62)(41,48)(45,64)(50,56)Polycyclic presentation:
2^7|b^2=g,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 | #13657, #13668, #13671, … |
| maximal normal subgroup of | #13657, #13668, #13671, … |
| maximal quotient of | #13616, #13618, #13634, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 996);