Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10,13,18,12,17,14,16)(11,19,22,24,21,15,20,23);(1,3,5,4)(2,6,7,8)(9,11,12,21)(10,15,17,19)(13,22,14,20)(16,24,18,23);(1,4,5,3)(2,6,7,8)(10,16)(11,20)(13,14)(15,19)(17,18)(21,22);(2,7)(6,8)(10,17)(15,19)(16,18)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #267982, #267984, #267998, … |
| maximal normal subgroup of | #267982, #267984, #267998, … |
| maximal quotient of | #275229, #275236, #275241, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17610);