Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | Aut(G) = C2×C8×D4 (order 128) · Inn(G) = D4 · Out(G) = C2×C8 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | D4 |
| commutator | C4 | C2×C16 |
| Frattini, cosocle | C2×C16 | C22 |
| socle | C2 | C22⋊C16 |
| prime radical | #41 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,6,12,24,37,7,13,25,38,27,40,52,60,8,14,26,39,28,41,53,61,4,10,21,34,22,35,49,58)(3,15,17,42,18,43,46,62,16,32,44,36,45,56,63,59,19,33,23,50,47,57,51,64,20,9,29,30,48,31,54,55);(1,3)(2,9)(4,20)(5,23)(6,18)(7,16)(8,19)(10,33)(11,36)(12,31)(13,15)(14,32)(17,26)(21,44)(22,48)(24,51)(25,29)(27,45)(28,47)(30,39)(34,42)(35,57)(37,59)(38,50)(40,43)(41,56)(46,53)(49,63)(52,54)(55,61)(58,62)(60,64)Polycyclic presentation:
2^7|a^2=d,c^2=g,d^2=e,e^2=f,f^2=g,b^a=bc,c^b=cg,d^b=dg56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1401, #13558, #13647, … |
| maximal normal subgroup of | C2×#1401, #13558, #13647, … |
| maximal quotient of | #7335, #7363, D4⋊C32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 133);