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) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
451 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 451 conjugacy classes of subgroups — too many to draw. Only its 147 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)(5,6,9,12)(7,11,8,10)(13,14,18,20,16,21,17,22)(15,23,24,27,25,19,26,28);(5,7,9,8)(6,10,12,11)(13,15,17,26,16,25,18,24)(14,19,22,27,21,23,20,28);(2,4)(14,20)(15,24)(17,18)(19,23)(21,22)(25,26);(5,8,9,7)(6,11,12,10)(14,21)(19,23)(20,22)(27,28)Polycyclic presentation:
2^8|a^2=ef,b^2=egh,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #501541, #501542, #501545, … |
| maximal normal subgroup of | #501541, #501542, #501545, … |
| maximal quotient of | #323268, #323276, #323284, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21522);