Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14)(15,22)(16,21)(17,27)(18,24)(19,23)(20,25)(26,28);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,15,18,16,20,26,19,17)(14,21,23,22,25,27,24,28);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,26,20,17,19,15)(14,22,23,27,25,28,24,21);(6,11)(10,12)(13,17,19,26,20,16,18,15)(14,22,24,21,25,28,23,27)Polycyclic presentation:
2^8|b^2=egh,c^2=e,d^2=f,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28915, #6459968, #6459987, … |
| maximal normal subgroup of | C2×#28915, #6459968, #6459987, … |
| maximal quotient of | #325995, #325996, #325999, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21903);