Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| 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 | #537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
319 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 319 conjugacy classes of subgroups — too many to draw. Only its 107 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)(3,4)(5,7)(6,8)(9,10,13,17,14,18,24,15)(11,12,19,21,16,23,20,22);(1,3)(2,6)(4,7)(5,8)(9,11,14,16)(10,12,18,23)(13,19,24,20)(15,22,17,21);(1,4,5,6)(2,3,7,8)(10,15)(11,19)(13,24)(16,20)(17,18)(21,22);(2,7)(3,8)(9,12)(10,16)(11,18)(13,22)(14,23)(15,20)(17,19)(21,24)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #151249, #151251, #151421, … |
| maximal normal subgroup of | #151249, #151251, #151421, … |
| maximal quotient of | #343697, #343698, #343700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24103);