Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| 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 | |
3572 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)(3,4)(5,6)(7,8)(9,10)(11,16)(12,15)(13,17)(14,19)(18,23)(20,21)(22,24);(2,6)(3,7)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24);(1,3,5,7)(2,4,6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(1,4)(2,7)(3,6)(5,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,23)(17,20)(19,24);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24);(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,21)(23,24)Polycyclic presentation:
2^8|c^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh70 conjugacy classes — view the full character table.
| maximal subgroup of | #347440, #347460, #347482, … |
| maximal normal subgroup of | #347440, #347460, #347482, … |
| maximal quotient of | #7627407, #7627409, #7627435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56037);