Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
576 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)(11,17)(12,14)(13,21)(15,19)(16,20)(18,24)(22,23);(1,2)(3,4)(5,6)(7,8)(10,14)(15,23)(16,21)(17,24);(6,8)(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17);(2,4)(10,16)(13,20)(14,21)(15,24)(17,23)(19,22)Polycyclic presentation:
2^8|a^2=eh,c^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=deh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6632028, #6632032, #6632042, … |
| maximal normal subgroup of | #6632028, #6632032, #6632042, … |
| maximal quotient of | #332939, #332940, #332941, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23501);