Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| 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 | C23 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
311 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 143 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)(9,10,14,19)(11,20,22,17)(12,21,23,24)(13,16,15,18);(10,16,19,18)(11,21,22,24)(12,23)(17,20);(1,3,4,8)(2,5,6,7)(9,11,14,22)(10,17,19,20)(12,18,23,16)(13,21,15,24);(9,12,13,17,14,23,15,20)(10,11,18,21,19,22,16,24)Polycyclic presentation:
2^8|a^2=f,b^2=eg,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,d^b=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #151312, #151313, #151450, … |
| maximal normal subgroup of | #151312, #151313, #151450, … |
| maximal quotient of | #325818, #325819, #325820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21814);