Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
366 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,7)(4,6)(5,8)(10,13)(11,15)(14,16);(2,6)(3,8)(4,7)(10,14)(13,16);(9,10,12,14)(11,16,15,13);(1,3,5,6)(2,7,8,4)Polycyclic presentation:
2^8|c^2=h,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=ch,d^b=de,e^b=eg,f^c=fh,e^d=eg37 conjugacy classes — view the full character table.
| maximal subgroup of | #153631, C2×#20084, #6359420, … |
| maximal normal subgroup of | #153631, C2×#20084, #6359420, … |
| maximal quotient of | #222345, #222565, #226063, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13072);