Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#493 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
341 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(10,16)(11,21)(12,23)(13,24)(14,17)(18,22)(19,20);(2,5)(6,8)(9,10,13,19,14,20,12,16,15,18,23,11,17,21,24,22);(1,3,4,7)(2,6,5,8)(9,11,12,10,17,16,13,21,15,19,24,18,14,22,23,20);(10,18)(11,19)(16,22)(20,21)Polycyclic presentation:
2^8|b^2=f,c^2=e,e^2=gh,f^2=g,g^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fgh,g^a=gh,d^b=dh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137518, #137574, #137577, … |
| maximal normal subgroup of | #137518, #137574, #137577, … |
| maximal quotient of | #494087, #494915, #495123, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26692);