Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | Aut(G) = C22×#7622880 (order 2048) · Inn(G) = #535 · Out(G) = C25 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2482 |
| prime radical | #585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
364 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,10)(3,8,14,23,15,24,21,28)(4,16,20,9,18,27,19,26)(13,25,17,32,29,31,30,22);(1,3)(2,8)(4,17,18,30)(5,21)(6,14)(7,15)(9,25,26,31)(10,28)(11,23)(12,24)(13,20,29,19)(16,22,27,32);(1,4)(2,9)(3,13)(5,20)(6,19)(7,18)(8,22)(10,27)(11,16)(12,26)(14,30)(15,29)(17,21)(23,31)(24,32)(25,28);(1,3)(2,8)(4,13)(5,21)(6,14)(7,15)(9,22)(10,28)(11,23)(12,24)(16,31)(17,20)(18,29)(19,30)(25,27)(26,32)Polycyclic presentation:
2^8|a^2=g,b^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fh,e^c=eh,f^c=fh,g^c=gh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #149893, #149894, #508104, … |
| maximal normal subgroup of | #149893, #149894, #508104, … |
| maximal quotient of | #314601, #314602, #314603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20586);