Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
491 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,9,12)(7,11,8,10)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,27)(25,28);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,14)(15,19)(16,26)(17,20)(18,21)(22,24)(23,28)(25,27);(2,4)(13,15,17,24,18,25,16,23)(14,19,20,22,21,27,26,28);(5,8,9,7)(6,11,12,10)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,27,28)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25822, #6445694, #6445701, … |
| maximal normal subgroup of | C2×#25822, #6445694, #6445701, … |
| maximal quotient of | #302133, #302134, #302135, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18810);