Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29634 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
679 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 679 conjugacy classes of subgroups — too many to draw. Only its 119 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)(11,15)(12,16)(13,17)(14,18)(19,24)(20,22)(21,23);(2,5)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,24)(17,22)(18,23);(1,3)(2,6)(4,7)(5,8)(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,23,24);(3,7)(6,8)(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,22,23,24)Polycyclic presentation:
2^8|c^2=h,d^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,f^b=fh,d^c=de37 conjugacy classes — view the full character table.
| maximal subgroup of | #246369, #246370, #246373, … |
| maximal normal subgroup of | #246369, #246370, #246373, … |
| maximal quotient of | #326725, #326726, #326727, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22625);