Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | Aut(G) = #72287333 (order 8192) · Inn(G) = D32 · Out(G) = D4×C16 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | D32 |
| commutator | C32 | C2×C4 |
| Frattini, cosocle | C2×C32 | C22 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
33 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 68 points:
(1,2,3,4)(6,12)(7,13)(8,18)(9,22)(10,25)(14,36)(15,40)(16,43)(17,31)(19,47)(20,50)(21,35)(23,52)(24,46)(26,51)(27,28)(29,37)(30,41)(32,33)(34,48)(38,65)(39,63)(42,66)(44,45)(49,68)(53,62)(54,61)(55,57)(56,67)(58,64)(59,60);(5,6,13,28,8,14,32,40,22,37,47,62,45,54,20,30,10,16,34,56,23,38,49,58,26,42,60,63,46,55,21,31,11,17,35,57,24,39,59,66,51,64,68,65,52,67,48,43,25,41,50,61,44,53,19,29,9,15,33,36,18,27,7,12)Polycyclic presentation:
2^8|a^2=d,b^2=ce,c^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | #94832, #94833, #94840, … |
| maximal normal subgroup of | #94832, #94833, #94840, … |
| maximal quotient of | #94749, #94761, #94840, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 533);