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) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#548 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,21)(16,27)(17,22)(18,24)(19,23)(20,25)(26,28);(5,7,9,8)(6,10,12,11)(13,15,17,25)(14,20,22,21)(16,19,26,18)(23,27,24,28);(1,2,3,4)(5,7,9,8)(6,11,12,10)(13,16,19,25,17,26,18,15)(14,21,24,28,22,20,23,27);(5,8,9,7)(6,11,12,10)(14,22)(20,21)(23,24)(27,28)Polycyclic presentation:
2^8|b^2=e,c^2=ef,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25889, #6446662, #6446673, … |
| maximal normal subgroup of | C2×#25889, #6446662, #6446673, … |
| maximal quotient of | #303213, #303223, #303234, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18877);