Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
711 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 711 conjugacy classes of subgroups — too many to draw. Only its 135 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) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,17)(16,23)(18,22)(19,26)(20,24)(21,27)(25,28);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,17,14)(16,21,25,19)(18,22,24,20)(23,26,28,27);(2,4)(5,7,9,8)(6,11,12,10)(13,16,24,27,17,25,18,26)(14,19,22,28,15,21,20,23);(5,8,9,7)(6,11,12,10)(14,15)(19,21)(20,22)(23,28)Polycyclic presentation:
2^8|b^2=e,c^2=ef,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #152783, #152784, #152786, … |
| maximal normal subgroup of | #152783, #152784, #152786, … |
| maximal quotient of | #303173, #303176, #303185, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18855);