Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23903 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1048 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 143 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)(7,10)(8,11)(9,13)(12,15)(14,18)(16,19)(17,20);(1,2)(3,4)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19);(2,4)(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20);(5,7)(6,12)(8,14)(9,15)(10,13)(11,18)(16,20)(17,19)Polycyclic presentation:
2^8|b^a=be,c^a=cf,d^a=deh,c^b=cg,f^b=fh,e^c=eh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139808, #501173, #501175, … |
| maximal normal subgroup of | #139808, #501173, #501175, … |
| maximal quotient of | #245246, #245251, #245254, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16891);