Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | Aut(G) order 8192 |
| 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 | #3499 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
668 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 668 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)(5,6,11,15)(7,12,17,8)(9,20,10,14)(13,19,16,18);(1,2,4,3)(5,7)(6,12)(8,15)(9,16)(10,13)(11,17)(14,19)(18,20);(5,8,11,12)(6,13,15,16)(7,14,17,20)(9,19,10,18);(1,3,4,2)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #250137, #250139, #250146, … |
| maximal normal subgroup of | #250137, #250139, #250146, … |
| maximal quotient of | #261272, #261273, #261284, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17221);