Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
481 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 481 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) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14,19,26,17,16,18,21)(15,22,27,23,24,20,25,28);(6,9)(10,11)(13,15,18,25,17,24,19,27)(14,20,21,23,16,22,26,28);(2,4)(13,16,17,14)(15,23,24,28)(18,21,19,26)(20,27,22,25);(5,7,8,12)(6,10,9,11)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #288775, #288786, #288803, … |
| maximal normal subgroup of | #288775, #288786, #288803, … |
| maximal quotient of | #320222, #320233, #320248, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21406);