Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,21)(20,24)(22,26)(23,25)(27,28);(6,9)(10,12)(13,15,17,22)(14,18,21,26)(16,23,24,28)(19,25,20,27);(5,7,8,11)(6,10,9,12)(14,19)(16,24)(18,25)(20,21)(23,28)(26,27);(1,3,4,2)(15,22)(18,26)(23,28)(25,27)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=e,d^2=g,f^2=h,b^a=be,c^a=cf,f^a=fh,d^b=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#21588, #6373048, #6373054, … |
| maximal normal subgroup of | C2×#21588, #6373048, #6373054, … |
| maximal quotient of | #212040, #212048, #212072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14576);