Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2549 conjugacy classes of subgroups.
| 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)(7,14)(8,13)(9,17)(10,12)(11,15)(16,20)(18,19);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,20)(14,17);(2,4);(6,12)(11,19)(13,17)(14,20);(5,8,10,9)(6,13,12,17)(7,15,16,18)(11,20,19,14)Polycyclic presentation:
2^8|e^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #114428, #114693, #114804, … |
| maximal normal subgroup of | #114428, #114693, #114804, … |
| maximal quotient of | #6346377, #6346383, #6346389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53347);