Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
473 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 473 conjugacy classes of subgroups — too many to draw. Only its 147 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,8,7)(9,10,12,15)(11,17,19,14)(13,16,20,23)(18,24,21,22);(1,2,4,3)(5,7,8,6)(11,18)(13,20)(14,22)(16,23)(17,24)(19,21);(6,7)(9,11,12,19)(10,14,15,17)(13,21,20,18)(16,24,23,22);(1,3,4,2)Polycyclic presentation:
2^8|a^2=f,b^2=efh,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #282565, #282582, #282595, … |
| maximal normal subgroup of | #282565, #282582, #282595, … |
| maximal quotient of | #287775, #287776, #287777, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18358);