Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6,8,10)(7,9,11,12)(13,14)(15,18)(16,19)(17,21)(20,24)(22,26)(23,25)(27,28);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,18)(16,23)(17,22)(19,25)(20,27)(21,26)(24,28);(7,11)(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=e,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27654, #6457203, #6457210, … |
| maximal normal subgroup of | C2×#27654, #6457203, #6457210, … |
| maximal quotient of | #314829, #314830, #314831, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20642);