Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
643 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 643 conjugacy classes of subgroups — too many to draw. Only its 147 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:
(1,2,4,6)(3,7,8,5)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,22,23,19);(1,3,4,8)(2,5,6,7)(10,14)(15,23)(16,21)(17,24);(9,11)(10,15)(12,18)(13,22)(14,23)(16,24)(17,21)(19,20);(13,20)(16,21)(17,24)(19,22)Polycyclic presentation:
2^8|a^2=e,b^2=eh,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,f^c=fh,g^c=gh,f^d=fh49 conjugacy classes — view the full character table.
| maximal subgroup of | #500479, #500480, #500481, … |
| maximal normal subgroup of | #500479, #500480, #500481, … |
| maximal quotient of | #349202, #349203, #349207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25961);