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 | |
3537 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)(11,17)(15,19)(18,23)(22,24);(1,2,5,4)(3,7,8,6)(9,10,14,13)(11,15,17,19)(12,16,20,21)(18,22,23,24);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24);(9,13,14,10)(11,19,17,15)(12,16,20,21)(18,22,23,24);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)Polycyclic presentation:
2^8|b^2=gh,c^2=g,d^2=g,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 | #347077, #347101, #347125, … |
| maximal normal subgroup of | #347077, #347101, #347125, … |
| maximal quotient of | #7626606, #7626638, #7626691, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55919);