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) = C27 |
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 | |
897 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)(11,18)(12,14)(13,17)(15,19)(16,21)(20,24)(22,23);(5,6)(7,8)(10,14)(12,21)(15,18)(16,17)(19,22)(23,24);(2,4)(6,8);(9,11,13,20)(10,15,17,23)(12,22,21,19)(14,18,16,24)Polycyclic presentation:
2^8|d^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×#23854, #6437496, #6437515, … |
| maximal normal subgroup of | C2×#23854, #6437496, #6437515, … |
| maximal quotient of | #244799, #244800, #244814, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16842);