Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3637 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,12)(13,21)(14,15)(16,17)(18,22)(19,23)(20,24);(2,5)(6,7)(10,15)(11,17)(18,21)(19,24);(1,2)(3,6)(4,5)(7,8)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20);(1,3,4,8)(2,6,5,7)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19);(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22);(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)Polycyclic presentation:
2^8|c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #185356, #185360, #185364, … |
| maximal normal subgroup of | #185356, #185360, #185364, … |
| maximal quotient of | #7626210, #7626219, #7626245, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55810);