Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| 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 | |
3651 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,7)(9,10)(11,18)(12,21)(13,16)(14,20)(15,17)(19,23)(22,24);(1,2,5,4)(3,7,8,6)(9,11)(10,14)(12,22)(13,19)(15,20)(16,24)(17,18)(21,23);(1,3,5,8)(2,6,4,7)(9,12,15,13)(10,16,17,21)(11,19,20,22)(14,23,18,24);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,19)(13,22)(15,20)(16,23)(17,18)(21,24);(9,13,15,12)(10,16,17,21)(11,19,20,22)(14,24,18,23);(9,14)(10,11)(12,23)(13,24)(15,18)(16,19)(17,20)(21,22)Polycyclic presentation:
2^8|b^2=gh,c^2=g,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh73 conjugacy classes — view the full character table.
| maximal subgroup of | #346797, #346821, #346845, … |
| maximal normal subgroup of | #346797, #346821, #346845, … |
| maximal quotient of | #7626626, #7626638, #7626711, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55949);