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 | #2414 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
562 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)(9,10,13,17,14,18,12,16)(11,15,22,24,20,23,21,19);(2,4)(5,6)(7,8)(9,11,12,21,14,20,13,22)(10,15,16,19,18,23,17,24);(6,8)(9,10,14,18)(11,19,20,24)(12,17,13,16)(15,22,23,21);(2,4)(11,20)(15,23)(19,24)(21,22)Polycyclic presentation:
2^8|a^2=fh,b^2=f,c^2=h,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×#25122, #6438933, #6438950, … |
| maximal normal subgroup of | C2×#25122, #6438933, #6438950, … |
| maximal quotient of | #281823, #281825, #281831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18110);