Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | Aut(G) = C2×#72311718 (order 4096) · Inn(G) = #481 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #481 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(7,13)(10,18)(14,20)(15,19);(6,10,11,18)(7,14)(8,16)(12,19,17,15)(13,20);(2,4)(5,6,8,17,9,11,16,12)(7,15,14,10,13,19,20,18)Polycyclic presentation:
2^8|a^2=e,b^2=d,c^2=dg,f^2=g,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,f^b=fg,g^b=gh,d^c=dh,f^c=fg,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #126351, #126352, #126353, … |
| maximal normal subgroup of | #126351, #126352, #126353, … |
| maximal quotient of | #146153, #146158, #146168, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6498);