Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | Aut(G) order 32768 |
| 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 |
| prime radical | #3151 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
527 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 527 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:
(5,6,10,15,11,8,17,14)(7,16,9,12,18,13,20,19);(1,2,3,4)(5,7,17,20,11,18,10,9)(6,12,15,13,8,19,14,16);(2,4)(5,8)(6,11)(7,12)(9,16)(10,15)(13,20)(14,17)(18,19);(5,9,10,18,11,20,17,7)(6,13,14,12,8,16,15,19)Polycyclic presentation:
2^8|a^2=f,b^2=eg,d^2=e,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 | #152939, #152942, #263819, … |
| maximal normal subgroup of | #152939, #152942, #263819, … |
| maximal quotient of | #321509, #321514, #321520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21482);