Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C8×S4 (order 384) · Inn(G) = C22 · Out(G) = C2×D3×C8 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C32 | C22 |
| commutator, socle | C2 | C22×C16 |
| Frattini, cosocle | C16 | C23 |
| prime radical | #41 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
53 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,9,5,10,17,27,6,11,18,28,20,30,39,49,7,12,19,29,21,31,40,50,22,32,41,51,42,52,57,62)(3,8,14,24,15,25,36,46,16,26,37,47,38,48,56,61,13,23,33,43,34,44,53,58,35,45,54,59,55,60,63,64);(1,3)(2,8)(4,14)(5,15)(6,16)(7,13)(9,24)(10,25)(11,26)(12,23)(17,36)(18,37)(19,33)(20,38)(21,34)(22,35)(27,46)(28,47)(29,43)(30,48)(31,44)(32,45)(39,56)(40,53)(41,54)(42,55)(49,61)(50,58)(51,59)(52,60)(57,63)(62,64);(3,13)(8,23)(14,33)(15,34)(16,35)(24,43)(25,44)(26,45)(36,53)(37,54)(38,55)(46,58)(47,59)(48,60)(56,63)(61,64)Polycyclic presentation:
2^7|a^2=d,d^2=e,e^2=f,f^2=g,c^b=cg80 conjugacy classes — view the full character table.
| Fitting subgroup of | #2258⋊C3, #11710971, #11711660, … |
| maximal subgroup of | #7514, #13647, #13648, … |
| maximal normal subgroup of | #7514, #13647, #13648, … |
| maximal quotient of | #13549, #13554, #13628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 990);