Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
383 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 383 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) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,9,12)(7,11,8,10);(5,7,9,8)(6,10,12,11)(13,14,18,26,17,21,16,20)(15,22,23,19,25,27,24,28);(2,4)(13,15,17,25)(14,19,21,28)(16,23,18,24)(20,27,26,22);(5,8,9,7)(6,11,12,10)(13,16,17,18)(14,20,21,26)(15,23,25,24)(19,27,28,22)Polycyclic presentation:
2^8|a^2=e,b^2=egh,c^2=h,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #143734, #143782, #145191, … |
| maximal normal subgroup of | #143734, #143782, #145191, … |
| maximal quotient of | #318704, #318705, #318706, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21256);