Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1375 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(5,6,9,7)(8,11,12,10)(13,14,17,16)(15,19,20,18);(1,3,4,2)(6,7)(10,11)(14,16)(18,19);(13,15)(14,18)(16,19)(17,20);(5,7,9,6)(8,11,12,10)(13,16,17,14)(15,19,20,18);(5,8,9,12)(6,10,7,11)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^8|a^2=h,b^2=fh,d^2=f,e^2=fg,b^a=bf,c^a=cg,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| maximal subgroup of | #107436, #107604, #107858, … |
| maximal normal subgroup of | #107436, #107604, #107858, … |
| maximal quotient of | #513567, #513586, #513601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27224);