Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
549 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 549 conjugacy classes of subgroups — too many to draw. Only its 167 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,14,10,12,9,13)(7,15,16,11,17,19,18,20);(1,2)(3,4)(5,7,8,16,10,17,9,18)(6,11,13,15,12,20,14,19);(2,4)(6,12)(7,16)(8,9)(11,19)(15,20)(17,18);(7,17)(11,20)(15,19)(16,18)Polycyclic presentation:
2^8|a^2=ef,b^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=dh,e^c=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #395737, #395906, #395950, … |
| maximal normal subgroup of | #395737, #395906, #395950, … |
| maximal quotient of | #242365, #242370, #242375, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16590);