Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C2×#483590 (order 1024) · Inn(G) = D16 · Out(G) = C22×C8 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | D16 |
| commutator | C16 | C2×C8 |
| Frattini, cosocle | C4×C16 | C22 |
| socle | C2 | #1331 |
| prime radical | #508 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,7,12,27,40,8,14,28,41,31,13,6,29)(3,15,17,43,18,44,49,63,20,45,50,64,19,51,30,36)(4,21,23,37,25,52,58,39,26,54,59,38,60,53,24,10)(9,22,32,46,34,55,61,48,35,57,62,47,42,56,33,16);(1,3)(2,9)(4,22)(5,17)(6,30)(7,18)(8,20)(10,36)(11,32)(12,34)(13,42)(14,35)(15,21)(16,24)(19,31)(23,46)(25,55)(26,57)(27,49)(28,50)(29,33)(37,43)(38,64)(39,63)(40,61)(41,62)(44,52)(45,54)(47,59)(48,58)(51,53)(56,60)Polycyclic presentation:
2^8|a^2=d,c^2=eg,d^2=f,e^2=gh,f^2=h,g^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=ce,e^b=eg,g^b=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7387, #150067, #150144, … |
| maximal normal subgroup of | C2×#7387, #150067, #150144, … |
| maximal quotient of | #93747, #93748, #93764, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 375);