Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| 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 | |
3674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,18)(12,20)(13,15)(14,17)(16,22)(19,23)(21,24);(1,2)(3,6)(4,7)(5,8)(9,11,14,13)(10,15,17,18)(12,19,22,21)(16,23,20,24);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24);(1,3)(2,6)(4,8)(5,7)(9,13,14,11)(10,15,17,18)(12,21,22,19)(16,23,20,24);(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,18)(14,17)(19,23)(20,22)(21,24)Polycyclic presentation:
2^8|b^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #493636, #493642, #511821, … |
| maximal normal subgroup of | #493636, #493642, #511821, … |
| maximal quotient of | #7627212, #7627217, #7627220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56005);