Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24464 |
| 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 | #535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
494 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 494 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)(9,10,13,17)(11,16,20,15)(12,22,21,24)(14,19,23,18);(2,5)(6,7)(9,11)(10,15)(12,23)(13,20)(14,21)(16,17)(18,24)(19,22);(9,12,14,11,13,21,23,20)(10,16,18,22,17,15,19,24);(1,3,4,8)(2,6,5,7)(10,17)(15,16)(18,19)(22,24)Polycyclic presentation:
2^8|a^2=e,c^2=f,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25689, #6450097, #6450104, … |
| maximal normal subgroup of | C2×#25689, #6450097, #6450104, … |
| maximal quotient of | #295479, #295484, #295489, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18677);