Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19883 |
| 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 | C42 | C24 |
| prime radical | C2×D8 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
312 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 103 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:
(2,5)(3,8)(6,7)(9,10,14,17)(11,15,13,21)(12,19,20,23)(16,22,24,18);(1,2)(3,7)(4,6)(5,8)(9,11,14,13)(10,15,17,21)(12,18,20,22)(16,23,24,19);(2,6)(5,7)(9,12,14,20)(10,16,17,24)(11,18,13,22)(15,23,21,19);(2,7)(3,8)(5,6)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=cg,d^b=de,e^b=eg,f^c=fh,e^d=eg43 conjugacy classes — view the full character table.
| maximal subgroup of | #491724, #491726, #491740, … |
| maximal normal subgroup of | #491724, #491726, #491740, … |
| maximal quotient of | #220784, #220956, #220977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12921);