Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10192 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #127 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
245 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 245 conjugacy classes of subgroups — too many to draw. Only its 63 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:
(2,7)(3,8)(4,5)(9,10,15,12)(11,19,23,20)(13,18,14,17)(16,22,24,21);(1,2,6,3)(4,8,5,7)(9,11,15,23)(10,16,12,24)(13,22,14,21)(17,20,18,19);(1,3,6,2)(4,8,5,7)(9,12,15,10)(11,20,23,19)(13,17,14,18)(16,21,24,22)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=g,d^2=g,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,c^b=cf,d^b=dg,f^b=fh,d^c=dh,e^c=egh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10203, #237384, #237385, … |
| maximal normal subgroup of | C2×#10203, #237384, #237385, … |
| maximal quotient of | #102716, #102721, #102728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3191);