Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C23×C8 |
| socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
363 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,20,23,21,19,22,24);(2,5)(3,7)(17,19,20,24,21,18,22,23);(1,3,4,7)(2,6,5,8)(10,13)(14,16);(2,5)(6,8)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=e,d^2=f,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ce88 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22341, #6376833, #6376837, … |
| maximal normal subgroup of | C2×#22341, #6376833, #6376837, … |
| maximal quotient of | #355329, #355630, #355658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15329);