Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| 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 | |
439 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)(7,11)(8,10)(9,12)(14,19)(15,22)(16,18)(20,27)(23,28)(24,26);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,14)(15,23)(16,25)(17,19)(18,21)(20,24)(22,27)(26,28);(7,12)(9,11)(13,15,18,22,17,24,16,26)(14,20,21,28,19,23,25,27);(6,10)(9,11)(13,16,17,18)(14,21,19,25)(15,22,24,26)(20,27,23,28)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22201, #6376171, #6376177, … |
| maximal normal subgroup of | C2×#22201, #6376171, #6376177, … |
| maximal quotient of | #218773, #218785, #218821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15189);