Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3497 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
683 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 683 conjugacy classes of subgroups — too many to draw. Only its 137 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:
(2,3)(5,6)(7,11)(8,16)(9,18)(10,14)(12,17)(13,19)(15,20);(1,2,4,3)(5,7,10,8)(6,11,14,16)(9,17,19,15)(12,13,20,18);(5,8,10,7)(6,12,14,20)(9,15,19,17)(11,18,16,13);(1,3,4,2)Polycyclic presentation:
2^8|b^2=eh,c^2=h,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139553, #139555, #139619, … |
| maximal normal subgroup of | #139553, #139555, #139619, … |
| maximal quotient of | #261048, #261051, #261056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17213);