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)(9,10,13,18,15,20,14,19)(11,16,22,17,12,23,21,24);(1,2,4,3)(5,6)(7,8)(9,11,13,22,15,12,14,21)(10,16,18,17,20,23,19,24);(6,8)(9,12,15,11)(10,17,20,24)(13,22,14,21)(16,19,23,18);(1,3,4,2)(9,13,15,14)(10,18,20,19)(11,21,12,22)(16,24,23,17)Polycyclic presentation:
2^8|a^2=f,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×#25399, #6444353, #6444360, … |
| maximal normal subgroup of | C2×#25399, #6444353, #6444360, … |
| maximal quotient of | #289440, #289442, #289448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18387);