Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
432 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 125 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)(7,8,11,10)(13,14)(15,21)(16,19)(17,20)(18,23)(22,26)(24,28)(25,27);(5,7,9,11)(6,10,12,8)(13,15)(14,18)(16,23)(17,22)(19,21)(20,28)(24,25)(26,27);(2,4)(5,8,9,10)(6,11,12,7)(14,19)(15,22)(17,25)(18,26)(21,28)(23,24);(2,4)(6,12)(7,11)(15,23)(18,21)(22,24)(26,28)Polycyclic presentation:
2^8|a^2=ef,b^2=eh,c^2=eh,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #125715, #125719, #125912, … |
| maximal normal subgroup of | #125715, #125719, #125912, … |
| maximal quotient of | #344990, #344991, #344992, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24178);