Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7389 |
| automorphism | Aut(G) = #72284874 (order 1024) · Inn(G) = #1333 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1333 |
| commutator | C16 | C2×C8 |
| Frattini, cosocle | #346 | C22 |
| prime radical | C22 | #361 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #346 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,7,13,29,41,8,15,6,30,28,14,32,12)(3,16,18,40,19,49,26,48,20,51,53,63,33,50,31,62)(4,21,23,45,24,46,60,44,25,58,27,10,39,42,52,43)(9,22,34,17,36,54,47,56,38,59,64,57,37,55,35,61);(1,3,8,20)(2,9,15,38)(4,22,25,59)(5,26,6,31)(7,33,28,19)(10,40,45,63)(11,35,30,47)(12,34,41,64)(13,37,14,36)(16,46,51,42)(17,52,57,60)(18,32,53,29)(21,50,58,49)(23,56,27,61)(24,55,39,54)(43,62,44,48)Polycyclic presentation:
2^8|a^2=d,b^2=h,c^2=eh,d^2=f,e^2=g,f^2=h,g^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=ceg,d^b=dg,e^b=egh,f^b=fh,g^b=gh,d^c=dh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7391, #150074, #150130, … |
| maximal normal subgroup of | C2×#7391, #150074, #150130, … |
| maximal quotient of | #93758, #93767, #93923, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 379);