Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | Aut(G) = #72316158 (order 4096) · Inn(G) = #466 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #466 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1942 |
| prime radical | C23 | C2×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,13,8,14,7,11)(3,15,19,32,20,37,17,9)(4,21,23,53,26,54,25,52)(6,28,27,39,29,38,22,12)(10,33,35,56,36,55,34,24)(16,42,46,61,47,62,44,51)(18,48,49,60,50,31,43,41)(30,45,40,58,57,64,59,63);(1,3,5,19,8,20,7,17)(2,9,13,15,14,32,11,37)(4,16,23,46,26,47,25,44)(6,18,27,49,29,50,22,43)(10,30,35,40,36,57,34,59)(12,31,28,41,39,48,38,60)(21,51,53,42,54,61,52,62)(24,45,33,58,56,64,55,63);(1,4,22,55,7,25,29,56,8,26,27,33,5,23,6,24)(2,10,28,53,13,35,39,54,14,36,38,52,11,34,12,21)(3,16,43,63,17,44,50,64,20,47,49,58,19,46,18,45)(9,30,41,42,15,40,48,61,32,57,60,62,37,59,31,51)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=eg,d^2=h,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #152440, #152445, #152447, … |
| maximal normal subgroup of | #152440, #152445, #152447, … |
| maximal quotient of | #140345, #140346, #140367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6161);