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 | |
3553 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)(11,18)(15,22)(19,23)(20,24);(2,6)(3,7)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24);(1,3,5,7)(2,4,6,8)(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24);(1,4)(2,7)(3,6)(5,8)(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24);(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22);(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)Polycyclic presentation:
2^8|b^2=h,c^2=g,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 | #346554, #346562, #346570, … |
| maximal normal subgroup of | #346554, #346562, #346570, … |
| maximal quotient of | #7626579, #7626582, #7626664, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55889);