Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2407 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,13,17,14,11,15,18)(12,20,24,16,23,21,22,19);(2,4)(5,6)(7,8)(9,11,15,17,14,10,13,18)(12,21,22,16,23,20,24,19);(6,8)(9,12)(10,16)(11,19)(13,22)(14,23)(15,24)(17,20)(18,21);(2,4)(9,13,14,15)(10,17,11,18)(12,22,23,24)(16,20,19,21)Polycyclic presentation:
2^8|a^2=fh,b^2=f,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25129, #6438924, #6438945, … |
| maximal normal subgroup of | C2×#25129, #6438924, #6438945, … |
| maximal quotient of | #282076, #282078, #282084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18117);