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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2407 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10)(11,19)(12,15)(13,22)(14,18)(16,20)(17,23)(21,24);(5,6)(7,8)(9,11,13,12,14,20,23,21)(10,15,17,19,18,24,22,16);(2,4)(6,8);(9,12,13,20,14,21,23,11)(10,16,17,15,18,19,22,24)Polycyclic presentation:
2^8|b^2=e,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24274, #6442754, #6442766, … |
| maximal normal subgroup of | C2×#24274, #6442754, #6442766, … |
| maximal quotient of | #262787, #262789, #262791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17262);