Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7447 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1361 |
| commutator | C2×C16 | C2×C4 |
| Frattini, cosocle | C4×C16 | C22 |
| prime radical | C22 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
63 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,28,14,30,40,8,15,29,13,7,31,6,12)(3,16,18,47,32,49,56,64,21,50,55,39,20,57,19,48)(4,22,24,10,38,41,51,43,27,59,62,44,26,42,25,58)(9,23,33,17,36,52,45,63,37,61,35,54,46,60,34,53);(1,3,8,21)(2,9,15,37)(4,23,27,61)(5,19,29,56)(6,18,30,55)(7,32,28,20)(10,39,44,47)(11,34,13,45)(12,33,40,35)(14,46,31,36)(16,41,50,42)(17,51,54,25)(22,57,59,49)(24,53,62,63)(26,52,38,60)(43,64,58,48)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=fgh,d^2=efgh,e^2=gh,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gh,c^b=cf,d^b=defg,e^b=eg,f^b=fgh,g^b=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7449, #150950, #150978, … |
| maximal normal subgroup of | C2×#7449, #150950, #150978, … |
| maximal quotient of | #93865, #93870, #93872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 437);