Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | C2×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1389 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)(3,4)(5,6,10,13,9,12,18,20)(7,14,16,17,15,11,19,8);(2,4)(5,7)(6,11)(8,13)(9,15)(10,16)(12,14)(17,20)(18,19);(5,8,10,14,9,17,18,11)(6,7,13,16,12,15,20,19);(6,12)(8,17)(11,14)(13,20);(6,12)(7,15)(13,20)(16,19)Polycyclic presentation:
2^8|a^2=g,c^2=g,g^2=h,b^a=bf,d^a=dh,e^a=eh,e^b=eh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #105761, #105763, #105779, … |
| maximal normal subgroup of | #105761, #105763, #105779, … |
| maximal quotient of | #6345693, #6345694, #6345695, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53132);