Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C26 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
2055 conjugacy classes of subgroups; only the 759 normal subgroups are shown.
This group has 2055 conjugacy classes of subgroups — too many to draw. Only its 759 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 — 5 generators on 20 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,19)(16,18)(17,20);(6,9)(10,11)(13,14,16,18)(15,17,20,19);(13,15)(14,17)(16,20)(18,19);(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20);(5,7)(6,10)(8,12)(9,11)(13,15)(14,17)(16,20)(18,19)Polycyclic presentation:
2^8|a^2=h,b^2=g,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #106326, #106380, #106570, … |
| maximal normal subgroup of | #106326, #106380, #106570, … |
| maximal quotient of | #513545, #513554, #513681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27164);