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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
684 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 684 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)(5,6,10,14)(7,11,17,20)(8,12,9,13)(15,18,16,19);(1,2,4,3)(5,7)(6,11)(8,16)(9,15)(10,17)(12,19)(13,18)(14,20);(6,12)(7,15)(8,9)(13,14)(16,17)(18,19);(1,3,4,2)Polycyclic presentation:
2^8|a^2=h,b^2=e,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 | #250135, #250141, #250144, … |
| maximal normal subgroup of | #250135, #250141, #250144, … |
| maximal quotient of | #261268, #261269, #261280, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17219);