Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13717 |
| automorphism | Aut(G) = #72285396 (order 2048) · Inn(G) = #2233 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2233 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | #2218 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ C2×M32 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
85 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7)(3,11)(4,14)(5,17)(8,20)(9,25)(10,26)(12,27)(13,31)(15,21)(16,22)(18,30)(19,23)(24,32)(28,29);(3,12)(7,20)(11,27)(13,22)(15,28)(21,29)(23,26)(25,32);(1,2,5,19,6,8,17,10)(3,13,15,25,12,22,28,32)(4,16,14,9,18,31,30,24)(7,21,26,11,20,29,23,27)Polycyclic presentation:
2^8|c^2=dg,e^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=ef,f^a=fg,g^a=gh,e^b=eh,d^c=dh,e^c=efh,f^c=fg,g^c=gh31 conjugacy classes — view the full character table.
| maximal subgroup of | #151490, #151492, #151493, … |
| maximal normal subgroup of | #151490, #151492, #151493, … |
| maximal quotient of | #152373, #152429, #152519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6705);