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 | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
615 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 615 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) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14,17,20)(15,22,24,19)(16,25,27,28)(18,23,26,21);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,17,24)(14,19,20,22)(16,21,27,23)(18,25,26,28);(2,4)(5,7,9,8)(6,11,12,10)(13,16,26,22,17,27,18,19)(14,15,23,28,20,24,21,25);(5,8,9,7)(6,11,12,10)(14,20)(15,24)(21,23)(25,28)Polycyclic presentation:
2^8|a^2=h,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 | #152782, #152785, #152788, … |
| maximal normal subgroup of | #152782, #152785, #152788, … |
| maximal quotient of | #303174, #303175, #303186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18856);