Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7225 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | H(3,ℤ/4) |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C2×#131 | C22 |
| socle | C2 | #1289 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,10,7,12,19,35)(3,13,15,37,17,39,40,56)(4,18,20,9,23,32,22,34)(6,25,24,36,26,54,48,11)(8,27,28,14,31,41,30,43)(16,45,44,60,46,64,61,38)(21,50,49,47,52,33,51,59)(29,42,57,62,58,63,53,55);(1,3,6,16,19,40,48,61,7,17,26,46,5,15,24,44)(2,8,11,29,10,30,25,53,12,31,36,58,35,28,54,57)(4,14,21,42,20,41,49,62,23,43,52,63,22,27,51,55)(9,13,33,38,18,37,47,45,34,39,50,60,32,56,59,64)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fgh,d^2=h,e^2=fgh,f^2=h,b^a=bc,c^a=cf,e^a=efgh,g^a=gh,d^b=dg,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93769, #93774, #93775, … |
| maximal normal subgroup of | #93769, #93774, #93775, … |
| maximal quotient of | #93018, #93019, #93020, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 217);