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 | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
621 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 621 conjugacy classes of subgroups — too many to draw. Only its 135 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)(3,6)(4,5)(7,8)(9,10,15,11)(12,20,22,24)(13,18,14,17)(16,23,19,21);(2,5)(6,8)(9,11,15,10)(12,19,22,16)(13,18,14,17)(20,21,24,23);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,23)(14,21)(15,22)(17,24)(18,20);(9,12)(10,16)(11,19)(13,21)(14,23)(15,22)(17,20)(18,24)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,f^c=fh,g^d=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #246337, #246338, #246339, … |
| maximal normal subgroup of | #246337, #246338, #246339, … |
| maximal quotient of | #349916, #349917, #349932, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25989);