Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10564 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
245 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 245 conjugacy classes of subgroups — too many to draw. Only its 57 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 — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,18)(11,16,20,24)(12,21,19,15)(13,23,22,17);(1,3)(2,5)(4,7)(6,8)(9,11,14,20)(10,15,18,21)(12,22,19,13)(16,17,24,23);(5,8)(9,12,13,11,14,19,22,20)(10,16,17,21,18,24,23,15)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=d,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fh,c^b=cf,d^b=dg,e^b=eh,f^b=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | #123227, #123228, #123229, … |
| maximal normal subgroup of | #123227, #123228, #123229, … |
| maximal quotient of | #104861, #104862, #104867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3554);