Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,17)(16,23)(18,21)(19,22)(20,25)(24,28)(26,27);(6,9)(10,11)(13,15,19,21)(14,17,22,18)(16,20,26,28)(23,25,27,24);(2,4)(13,16,19,26)(14,20,22,28)(15,23,21,27)(17,24,18,25);(5,7,8,12)(6,10,9,11)(13,17,19,18)(14,15,22,21)(16,24,26,25)(20,23,28,27)Polycyclic presentation:
2^8|b^2=h,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24195, #6448789, #6448806, … |
| maximal normal subgroup of | C2×#24195, #6448789, #6448806, … |
| maximal quotient of | #258878, #258890, #258901, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17183);