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×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
471 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 471 conjugacy classes of subgroups — too many to draw. Only its 167 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:
(1,2,3,4)(5,6,9,12)(7,11,8,10)(13,14,17,15,18,20,16,19);(5,7,9,8)(6,10,12,11)(13,15,16,14,18,19,17,20);(2,4)(14,15)(16,17)(19,20);(5,8,9,7)(6,11,12,10)Polycyclic presentation:
2^8|a^2=efh,b^2=egh,d^2=e,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 | #288721, #288727, #288829, … |
| maximal normal subgroup of | #288721, #288727, #288829, … |
| maximal quotient of | #323264, #323271, #323278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21518);