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) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,23,21,19,20,24);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(17,19,20,23,21,18,22,24);(10,14)(13,16)(17,19,21,18)(20,24,22,23);(1,3,4,8)(2,6,5,7)(17,20,21,22)(18,23,19,24)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=h,d^2=e,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×#27741, #6458437, #6458442, … |
| maximal normal subgroup of | C2×#27741, #6458437, #6458442, … |
| maximal quotient of | #360406, #362568, #366065, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20729);