Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,12)(8,18)(9,14)(10,15)(11,16)(13,19)(17,20);(1,2,4,3)(5,7)(6,12)(8,16)(9,17)(10,13)(11,18)(14,20)(15,19);(5,8,11,12)(6,13,16,17)(7,15,18,14)(9,19,10,20);(1,3,4,2)(5,9,11,10)(6,14,16,15)(7,17,18,13)(8,19,12,20)Polycyclic presentation:
2^8|b^2=e,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24235, #6442604, #6442606, … |
| maximal normal subgroup of | C2×#24235, #6442604, #6442606, … |
| maximal quotient of | #261277, #261278, #261289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17223);