Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| 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 | |
337 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 117 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10,13,18,14,12,15,17)(11,19,21,20,22,16,23,24);(2,5)(6,7)(9,11)(10,16)(12,19)(13,21)(14,22)(15,23)(17,20)(18,24);(9,12,14,10)(11,20,22,24)(13,18,15,17)(16,21,19,23);(1,3,4,8)(2,6,5,7)(9,13,14,15)(10,17,12,18)(11,21,22,23)(16,20,19,24)Polycyclic presentation:
2^8|a^2=ef,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #272168, #272170, #272183, … |
| maximal normal subgroup of | #272168, #272170, #272183, … |
| maximal quotient of | #276381, #276388, #276393, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17710);