Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1305 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(6,10)(11,12)(13,14)(15,18)(16,20)(17,19);(5,6,9,10)(7,11,8,12)(13,15,18,14)(16,20,17,19);(5,7,9,8)(6,11,10,12)(13,16,18,17)(14,19,15,20);(5,7,9,8)(6,12,10,11)(13,17,18,16)(14,19,15,20);(5,8,9,7)(6,12,10,11)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^b=df100 conjugacy classes — view the full character table.
| Fitting subgroup of | #34129⋊C3 |
| maximal subgroup of | #105653, #106867, #107053, … |
| maximal normal subgroup of | #105653, #106867, #107053, … |
| maximal quotient of | #513469, #513480, #513499, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27117);