Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3486 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
869 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 869 conjugacy classes of subgroups — too many to draw. Only its 147 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:
(5,6,10,8)(7,13,15,20)(9,17,19,12)(11,18,14,16);(1,2)(3,4)(5,6,10,8)(7,11,15,14)(9,17,19,12)(13,16,20,18);(2,4)(5,7)(6,11)(8,14)(9,18)(10,15)(12,20)(13,17)(16,19);(5,8,10,6)(7,14,15,11)(9,12,19,17)(13,18,20,16)Polycyclic presentation:
2^8|a^2=h,b^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,f^b=fh,e^c=eh,f^c=fh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139711, #139762, #139784, … |
| maximal normal subgroup of | #139711, #139762, #139784, … |
| maximal quotient of | #245152, #245156, #245161, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16899);