Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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×#538 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1157 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(5,6,9,11)(7,8,12,10)(14,17)(15,16);(5,7,9,12)(6,10,11,8)(13,14,18,17)(15,20,16,19);(1,3,4,2)(13,15,18,16)(14,19,17,20);(5,7)(6,8)(9,12)(10,11)(13,16,18,15)(14,20,17,19);(5,8)(6,7)(9,10)(11,12)(13,17,18,14)(15,20,16,19)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=fh,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#34451, #7626015, #7626054, … |
| maximal normal subgroup of | C2×#34451, #7626015, #7626054, … |
| maximal quotient of | #513817, #513821, #513885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27439);