Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29742 |
| automorphism | Aut(G) order 32768 |
| 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×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
428 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 428 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,9)(7,10,11,12)(13,14,17,19,18,15,16,20);(1,2)(3,4)(6,9)(10,12);(2,4)(7,11)(10,12)(13,15)(14,18)(16,20)(17,19);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,19,15,20)Polycyclic presentation:
2^8|a^2=ef,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #268214, #268223, #268227, … |
| maximal normal subgroup of | #268214, #268223, #268227, … |
| maximal quotient of | #330473, #330477, #330481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22841);