Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
502 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 502 conjugacy classes of subgroups — too many to draw. Only its 137 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:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,20)(13,19)(15,17)(16,24)(18,22)(21,23);(1,3,5,4)(2,6,8,7)(9,11,14,10)(12,21,23,20)(13,19,15,17)(16,22,18,24);(9,12,14,23)(10,16,11,18)(13,22,15,24)(17,21,19,20);(1,4,5,3)(2,7,8,6)(9,13,14,15)(10,17,11,19)(12,22,23,24)(16,21,18,20)Polycyclic presentation:
2^8|a^2=eh,b^2=e,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #250105, #250107, #250114, … |
| maximal normal subgroup of | #250105, #250107, #250114, … |
| maximal quotient of | #265648, #265649, #265660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17311);