Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(6,7)(10,11)(13,14,16,18)(15,17,19,20)(21,22)(23,24)(25,28)(26,27);(5,6,9,7)(8,10,12,11)(13,15,16,19)(14,17,18,20)(21,23)(22,26)(24,28)(25,27);(1,2,3,4)(5,7,9,6)(8,10,12,11)(13,15)(14,17)(16,19)(18,20)(21,23)(22,24)(25,27)(26,28);(5,8)(6,10)(7,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,24)(22,27)(23,28)(25,26)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #291239, #291255, #291271, … |
| maximal normal subgroup of | #291239, #291255, #291271, … |
| maximal quotient of | #155686, #155800, #155801, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9686);