Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
345 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,13,15,16)(18,21)(19,22)(23,24);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(17,18,22,24,20,23,19,21);(2,6)(5,8)(11,15)(13,16);(17,18,22,24,20,23,19,21)Polycyclic presentation:
2^8|a^2=fg,b^2=efh,d^2=eh,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | #504928, #504932, #504937, … |
| maximal normal subgroup of | #504928, #504932, #504937, … |
| maximal quotient of | #360469, #362673, #365301, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20827);