Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,10,16,12,9,11,15)(7,17,19,14,8,20,18,13);(1,2)(3,4)(5,7,10,19,12,8,11,18)(6,13,16,17,9,14,15,20);(2,4)(5,8,12,7)(6,14,9,13)(10,19,11,18)(15,20,16,17);(5,9)(6,12)(7,14)(8,13)(10,15)(11,16)(17,18)(19,20)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,f^c=fh,g^c=gh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25190, #6439019, #6439026, … |
| maximal normal subgroup of | C2×#25190, #6439019, #6439026, … |
| maximal quotient of | #283583, #283585, #283591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18178);