Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| 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, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(6,9)(7,11)(13,14)(15,21)(16,18)(17,25)(19,23)(20,24)(22,28)(26,27);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(14,18)(19,27)(20,25)(21,28);(6,9)(10,12)(13,15,16,22)(14,19,18,27)(17,26,24,23)(20,28,25,21);(5,7,8,11)(6,10,9,12)(14,18)(19,27)(20,25)(21,28)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ceh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22121, #6375729, #6375740, … |
| maximal normal subgroup of | C2×#22121, #6375729, #6375740, … |
| maximal quotient of | #218181, #218187, #218205, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15109);