Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
553 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 553 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,10,15,11,8,18,14)(7,16,9,12,17,13,20,19);(1,2,3,4)(5,7,10,9,11,17,18,20)(6,12,14,16,8,19,15,13);(2,4)(5,8)(6,11)(7,12)(9,16)(10,15)(13,20)(14,18)(17,19);(5,9,18,7,11,20,10,17)(6,13,15,19,8,16,14,12)Polycyclic presentation:
2^8|a^2=f,b^2=egh,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139400, #139403, #143209, … |
| maximal normal subgroup of | #139400, #139403, #143209, … |
| maximal quotient of | #321512, #321518, #321521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21485);