Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| 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 |
| prime radical | #2407 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
878 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)(13,23)(14,18)(16,21)(17,20)(22,24);(5,6)(7,8)(9,11,14,21)(10,15,18,24)(12,13,22,20)(16,23,19,17);(2,4)(6,8);(9,12,14,22)(10,16,18,19)(11,20,21,13)(15,17,24,23)Polycyclic presentation:
2^8|b^2=h,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23868, #6437509, #6437513, … |
| maximal normal subgroup of | C2×#23868, #6437509, #6437513, … |
| maximal quotient of | #244937, #244938, #244953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16856);