Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C2×#72223646 (order 2048) · Inn(G) = D16 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | D16 |
| commutator | C16 | C2×C8 |
| Frattini, cosocle | C4×C16 | C22 |
| socle | C2 | #1371 |
| prime radical | C22 | #368 |
| 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 | |
39 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,7,12,26,38,8,14,27,39,29,13,6,28)(3,15,18,41,20,42,50,62,21,44,51,63,53,43,19,52)(4,22,23,35,24,54,47,37,25,56,57,36,46,55,16,10)(9,30,31,48,32,58,60,49,34,59,61,64,33,45,40,17);(1,3,16,45,6,19,46,64,29,53,57,59,27,51,25,49,8,21,47,58,26,50,24,48,7,20,23,30,5,18,4,17)(2,9,22,41,11,31,35,42,12,32,54,62,38,60,37,44,14,34,56,63,39,61,36,43,13,33,55,52,28,40,10,15)Polycyclic presentation:
2^8|a^2=d,b^2=ce,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=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #94611, #94612, #94737, … |
| maximal normal subgroup of | #94611, #94612, #94737, … |
| maximal quotient of | #93880, #93881, #94027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 450);