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 | |
3632 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)(3,5)(4,8)(6,7)(10,14)(12,19)(15,20)(18,23);(2,7)(3,8)(11,17)(15,20)(18,23)(22,24);(1,3,6,8)(2,4,7,5)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24);(1,4,6,5)(2,3,7,8);(1,2)(3,4)(5,8)(6,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(1,5,6,4)(2,8,7,3)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)Polycyclic presentation:
2^8|c^2=h,d^2=g,f^2=g,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 | #185423, #185429, #185435, … |
| maximal normal subgroup of | #185423, #185429, #185435, … |
| maximal quotient of | #7626217, #7626221, #7626227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55818);