Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| 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 |
| prime radical | #2407 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
593 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)(9,10,15,17,14,11,13,18)(12,16,22,23,21,19,20,24);(5,6)(7,8)(9,11,14,10)(12,19,21,16)(13,18,15,17)(20,24,22,23);(2,4)(6,8)(9,12,14,21)(10,16,11,19)(13,20,15,22)(17,23,18,24);(2,4)(10,17)(11,18)(13,15)(16,23)(19,24)(20,22)Polycyclic presentation:
2^8|a^2=efh,b^2=h,c^2=h,e^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6634509, #6634511, #6634513, … |
| maximal normal subgroup of | #6634509, #6634511, #6634513, … |
| maximal quotient of | #343014, #343015, #343016, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24049);