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 | #2431 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
567 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)(9,10,15,16,14,11,12,18)(13,21,20,19,23,24,22,17);(5,6)(7,8)(9,11,14,10)(12,18,15,16)(13,21,23,24)(17,22,19,20);(2,4)(6,8)(9,12,14,15)(10,16,11,18)(13,20,23,22)(17,24,19,21);(2,4)(9,13,14,23)(10,17,11,19)(12,20,15,22)(16,24,18,21)Polycyclic presentation:
2^8|a^2=eh,b^2=h,c^2=h,d^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 | #6632128, #6632129, #6632160, … |
| maximal normal subgroup of | #6632128, #6632129, #6632160, … |
| maximal quotient of | #333286, #333301, #333315, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23447);