Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24805 |
| automorphism | Aut(G) order 16384 |
| 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×#127 | C22 |
| 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 115 normal subgroups are shown.
This group has 337 conjugacy classes of subgroups — too many to draw. Only its 115 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)(11,19)(12,21)(13,17)(14,24)(15,18)(16,20)(22,23);(2,5)(6,7)(9,11,15,22,13,20,14,21)(10,16,18,12,17,19,24,23);(9,12,14,19,13,23,15,16)(10,11,18,21,17,20,24,22);(1,3,4,8)(2,6,5,7)(10,17)(11,20)(18,24)(21,22)Polycyclic presentation:
2^8|a^2=eh,b^2=g,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,d^b=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28674, #6464093, #6464094, … |
| maximal normal subgroup of | C2×#28674, #6464093, #6464094, … |
| maximal quotient of | #324828, #324831, #324835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21662);