Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 262144 |
| 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 | C2×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1319 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)(7,10)(8,11)(9,12)(13,14)(15,18)(16,17)(19,20);(1,3,4,2)(5,7,10,6)(8,11,9,12);(5,8,10,9)(6,11,7,12)(14,17)(18,20);(5,9,10,8)(6,11,7,12);(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^8|a^2=h,b^2=fh,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=cf88 conjugacy classes — view the full character table.
| maximal subgroup of | #106195, #107852, #108046, … |
| maximal normal subgroup of | #106195, #107852, #108046, … |
| maximal quotient of | #513390, #513402, #513412, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27074);