Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
345 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 345 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:
(1,2,4,6)(3,7,8,5)(9,10)(11,18)(12,22)(13,16)(14,17)(15,19)(20,23)(21,24);(9,10,13,22,14,17,12,16)(11,15,20,24,21,23,19,18);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,22)(14,17)(18,19)(20,24)(21,23);(9,11,12,19,14,21,13,20)(10,15,16,18,17,23,22,24)Polycyclic presentation:
2^8|a^2=f,b^2=eh,c^2=f,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #503659, #504129, #510186, … |
| maximal normal subgroup of | #503659, #504129, #510186, … |
| maximal quotient of | #305770, #305771, #305778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19069);