Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| 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 | #2540 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(9,10,13,18,14,19,12,17)(11,15,20,16,22,24,21,23);(1,2,4,3)(5,6)(7,8)(9,11,12,21,14,22,13,20)(10,15,17,23,19,24,18,16);(6,8)(9,11,14,22)(10,16,19,23)(12,20,13,21)(15,17,24,18);(1,3,4,2)(9,12,14,13)(10,17,19,18)(11,20,22,21)(15,16,24,23)Polycyclic presentation:
2^8|a^2=fh,b^2=ef,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25403, #6444353, #6444360, … |
| maximal normal subgroup of | C2×#25403, #6444353, #6444360, … |
| maximal quotient of | #289472, #289474, #289480, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18391);