Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| 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 | #3512 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
486 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 486 conjugacy classes of subgroups — too many to draw. Only its 123 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:
(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,16)(13,21)(14,17)(18,19)(20,23)(22,24);(2,5)(6,7)(9,11)(10,15)(12,19)(13,22)(14,20)(16,18)(17,23)(21,24);(9,10)(11,18,20,24)(12,21)(13,16)(14,17)(15,22,23,19);(1,3,4,8)(2,6,5,7)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,18,23,24)Polycyclic presentation:
2^8|a^2=e,c^2=f,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^b=gh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #268286, #268289, #268301, … |
| maximal normal subgroup of | #268286, #268289, #268301, … |
| maximal quotient of | #295333, #295334, #295337, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18658);