Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
427 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 427 conjugacy classes of subgroups — too many to draw. Only its 143 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:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,16,18)(15,21,22,28)(17,25,24,20)(19,26,27,23);(1,2,4,3)(6,9)(10,12)(14,18)(19,27)(20,25)(21,28);(1,3,4,2)(5,7,8,11)(6,10,9,12)(14,18)(19,27)(20,25)(21,28);(13,15,16,22)(14,19,18,27)(17,26,24,23)(20,28,25,21)Polycyclic presentation:
2^8|a^2=h,b^2=fh,c^2=eh,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,g^a=gh,g^d=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #100276, #100277, #100292, … |
| maximal normal subgroup of | #100276, #100277, #100292, … |
| maximal quotient of | #350913, #350915, #350917, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26021);