Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,22)(21,24)(23,27)(26,31)(28,30)(29,32);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(18,22)(19,26)(21,28)(23,31)(25,32)(27,29);(10,14)(13,16)(17,19,20,27)(18,23,22,25)(21,29,28,26)(24,31,30,32);(1,4,5,3)(2,7,8,6)(18,22)(19,27)(24,30)(26,29)Polycyclic presentation:
2^8|a^2=eh,b^2=ef,c^2=h,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25075, #6446013, #6446021, … |
| maximal normal subgroup of | C2×#25075, #6446013, #6446021, … |
| maximal quotient of | #360443, #363071, #364875, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18063);