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 | |
3574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,5)(6,8)(9,10,15,13)(11,18,16,12)(14,21,22,17)(19,20,23,24);(3,7)(6,8)(10,13)(11,16)(17,21)(19,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,18)(14,19)(15,16)(17,20)(21,24)(22,23);(9,12,15,18)(10,16,13,11)(14,20,22,24)(17,23,21,19);(1,3,4,7)(2,6,5,8)(9,13,15,10)(11,18,16,12)(14,21,22,17)(19,24,23,20);(1,2)(3,6)(4,5)(7,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)Polycyclic presentation:
2^8|a^2=g,d^2=g,e^2=h,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 | #185425, #185427, #185431, … |
| maximal normal subgroup of | #185425, #185427, #185431, … |
| maximal quotient of | #7626232, #7626234, #7626240, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55823);