Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | Aut(G) = #72231730 (order 2048) · Inn(G) = C2×D8 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24 |
| socle | C2 | C22×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,10)(5,17)(6,21)(7,9)(8,14)(11,19)(13,29)(15,28)(16,26)(18,24)(20,31)(22,27)(23,30)(25,32);(1,3)(2,8)(4,13)(5,18)(6,22)(7,15)(9,23)(10,24)(11,27)(12,17)(14,19)(16,30)(20,32)(21,25)(26,29)(28,31);(2,9)(8,23)(10,21)(11,26)(12,28)(17,31)(24,25)(27,29);(1,4,7,6)(2,10,9,21)(3,13,15,22)(5,16,20,19)(8,24,23,25)(11,17,26,31)(12,29,28,27)(14,18,30,32)Polycyclic presentation:
2^7|d^2=g,e^2=fg,f^2=g,b^a=be,c^a=cg,e^a=ef,f^a=fg,e^b=ef,f^b=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #13021, #13024, #13027, … |
| maximal normal subgroup of | #13021, #13024, #13027, … |
| maximal quotient of | #33380, #33409, #33451, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2146);