Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 403 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) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,11)(7,9)(8,15)(12,20)(13,19)(14,17)(16,18);(1,2,3,4);(5,6)(7,12)(8,16)(9,18)(10,13)(11,15)(14,19)(17,20);(5,7,10,9)(6,12,13,18)(8,14,17,15)(11,16,19,20)Polycyclic presentation:
2^8|b^2=e,d^2=h,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fg,g^a=gh,f^c=fg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136905, #136909, #136938, … |
| maximal normal subgroup of | #136905, #136909, #136938, … |
| maximal quotient of | #490430, #490455, #490496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26570);