Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C22×C42 |
| socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,12)(9,10)(13,14,17,16,18,20,25,24)(15,19,23,22,21,26,28,27);(1,2,4,3)(6,10)(11,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28);(1,3,4,2)(5,7,9,8)(6,11,10,12)(15,21)(19,26)(22,27)(23,28);(5,8,9,7)(6,12,10,11)(13,16,17,20,18,24,25,14)(15,22,23,26,21,27,28,19)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=eg,d^2=ef,f^2=h,b^a=be,d^a=dh,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #101472, #101474, #101476, … |
| maximal normal subgroup of | #101472, #101474, #101476, … |
| maximal quotient of | #184375, #184376, #184377, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13506);