Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| 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, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
369 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)(3,4)(5,6,8,9)(7,11,12,10);(6,9)(10,11)(13,14,16,20,17,19,25,27)(15,21,22,26,23,28,24,18);(2,4)(13,15,17,23)(14,18,19,26)(16,24,25,22)(20,28,27,21);(5,7,8,12)(6,10,9,11)(14,19)(18,26)(20,27)(21,28)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28198, #6463771, #6463779, … |
| maximal normal subgroup of | C2×#28198, #6463771, #6463779, … |
| maximal quotient of | #317759, #317760, #317761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21186);