Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator | C4 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | C2×#585 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
2451 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)(6,11)(8,9)(12,18)(13,17)(15,16)(19,20);(5,6)(7,12)(8,11)(9,17)(10,13)(14,19)(15,18)(16,20);(5,7)(6,12)(8,15)(9,16)(10,14)(11,18)(13,19)(17,20);(7,14)(12,19)(15,16)(18,20);(5,8,10,9)(6,11,13,17)(7,15,14,16)(12,18,19,20)Polycyclic presentation:
2^8|a^2=g,e^2=h,f^2=h,b^a=bf,e^a=eh,f^a=fh,f^b=fh,d^c=dh82 conjugacy classes — view the full character table.
| Fitting subgroup of | #60089⋊C3 |
| maximal subgroup of | #359683, #359684, #360591, … |
| maximal normal subgroup of | #359683, #359684, #360591, … |
| maximal quotient of | #6342910, #6342930, #6342936, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53077);