Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14)(15,17)(16,19)(18,20);(5,6,8,10)(7,9,12,11)(14,15)(18,19);(13,14,17,15)(16,18,20,19);(1,3,4,2)(5,6)(7,9)(8,10)(11,12)(13,15,17,14)(16,19,20,18);(5,7,8,12)(6,9,10,11)(13,16,17,20)(14,18,15,19)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=g,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | #96981, #129577, #209699, … |
| maximal normal subgroup of | #96981, #129577, #209699, … |
| maximal quotient of | #513833, #513847, #513914, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27486);