Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | #392 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
513 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)(5,6,8,7)(9,10,14,16,13,17,12,20)(11,15,18,24,19,23,21,22);(6,7)(9,10,12,20,13,17,14,16)(11,15,21,22,19,23,18,24);(2,4)(9,11)(10,15)(12,21)(13,19)(14,18)(16,24)(17,23)(20,22);(5,7,8,6)(9,12,13,14)(10,16,17,20)(11,18,19,21)(15,22,23,24)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,d^b=de,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#22992, #6381806, #6381810, … |
| maximal normal subgroup of | C2×#22992, #6381806, #6381810, … |
| maximal quotient of | #229750, #229751, #229752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15980);