Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3036 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
458 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 125 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:
(2,3)(5,6,10,12,9,14,8,13)(7,11,17,20,15,19,16,18);(1,2,4,3)(5,7,8,16,9,15,10,17)(6,11,13,18,14,19,12,20);(6,12)(7,15)(8,10)(11,18)(13,14)(19,20);(1,3,4,2)Polycyclic presentation:
2^8|a^2=fh,b^2=ef,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #285179, #285181, #285196, … |
| maximal normal subgroup of | #285179, #285181, #285196, … |
| maximal quotient of | #287740, #287745, #287752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18393);