Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(9,10)(11,18)(12,15)(13,17)(14,19)(16,22)(20,24)(21,23);(5,6)(7,8)(9,11,13,20)(10,14,17,23)(12,21,22,19)(15,18,16,24);(2,4)(6,8);(10,15)(12,22)(14,18)(16,17)(19,21)(23,24)Polycyclic presentation:
2^8|a^2=f,b^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24428, #6437820, #6437844, … |
| maximal normal subgroup of | C2×#24428, #6437820, #6437844, … |
| maximal quotient of | #268357, #268373, #268386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17416);