Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19230 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
320 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 121 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:
(1,2,4,7)(3,5,8,6)(9,10,14,18)(11,19,21,22)(12,16,15,24)(13,20,23,17);(9,11,13,16,14,21,23,24)(10,15,17,19,18,12,20,22);(2,5)(3,8)(6,7)(9,12)(10,16)(11,20)(13,22)(14,15)(17,21)(18,24)(19,23);(11,21)(12,15)(16,24)(19,22)Polycyclic presentation:
2^8|a^2=gh,b^2=e,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dg,d^c=dg,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #283591, #283592, #283609, … |
| maximal normal subgroup of | #283591, #283592, #283609, … |
| maximal quotient of | #189088, #197744, #197746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12221);