Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| 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 | |
349 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 349 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:
(2,4)(5,6,9,12)(7,11,8,10)(13,14)(15,18)(16,20)(17,19)(21,25)(22,27)(23,26)(24,28);(5,7,9,8)(6,10,12,11)(13,15,16,22,17,23,24,21)(14,18,20,27,19,26,28,25);(1,2,3,4)(15,21)(16,24)(18,25)(20,28)(22,23)(26,27);(5,8,9,7)(6,11,12,10)(14,19)(18,26)(20,28)(25,27)Polycyclic presentation:
2^8|a^2=e,b^2=eg,c^2=f,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #143290, #143291, #143391, … |
| maximal normal subgroup of | #143290, #143291, #143391, … |
| maximal quotient of | #325189, #325190, #325191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21668);