Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,11)(8,10)(9,12)(13,14,16,18)(15,17,19,20);(2,3)(5,6,8,10)(7,9,12,11);(1,2,4,3)(13,14)(15,17)(16,18)(19,20);(5,7)(6,9)(8,12)(10,11);(1,3,4,2)(5,6)(7,9)(8,10)(11,12)(13,15,16,19)(14,17,18,20)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=h,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #97736, #261316, #261320, … |
| maximal normal subgroup of | #97736, #261316, #261320, … |
| maximal quotient of | #521420, #522137, #522699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32409);