Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13709 |
| automorphism | Aut(G) = #72319682 (order 2048) · Inn(G) = #2226 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2226 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | #508 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,9)(3,13,16,31)(4,7)(5,18)(6,12)(10,26,15,32)(11,20)(14,27,23,29)(17,28)(19,22,30,24)(21,25);(1,2,7,6,8,12,4,9)(3,10,23,22,16,15,14,24)(5,11,20,18,21,28,17,25)(13,19,27,32,31,30,29,26);(1,3)(2,10)(4,14)(5,19)(6,22)(7,23)(8,16)(9,24)(11,27)(12,15)(13,25)(17,26)(18,31)(20,32)(21,30)(28,29)Polycyclic presentation:
2^8|a^2=dg,b^2=dh,d^2=h,e^2=fgh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ef,f^a=fg,g^a=gh,e^c=efh,f^c=fg,g^c=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | #151626, #151627, #151658, … |
| maximal normal subgroup of | #151626, #151627, #151658, … |
| maximal quotient of | #152282, #152448, #152802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6698);