Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1097 conjugacy classes of subgroups; only the 575 normal subgroups are shown.
This group has 1097 conjugacy classes of subgroups — too many to draw. Only its 575 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) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,9,12)(7,8,11,10)(13,14,16,18)(15,19,20,17);(1,2,3,4)(7,11)(8,10);(5,7,9,11)(6,10,12,8)(13,15,16,20)(14,17,18,19);(5,8,9,10)(6,11,12,7);(5,6)(7,10)(8,11)(9,12)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=g,d^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #97140, #111264, #111305, … |
| maximal normal subgroup of | #97140, #111264, #111305, … |
| maximal quotient of | #516459, #516636, #516880, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 28161);