Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,9)(7,10,11,12)(13,14,17,22,19,21,18,23)(15,24,25,28,16,27,26,20);(1,2)(3,4)(6,9)(10,12)(13,15,19,16)(14,20,21,28)(17,26,18,25)(22,27,23,24);(2,4)(13,16,19,15)(14,20,21,28)(17,25,18,26)(22,27,23,24);(5,7,8,11)(6,10,9,12)(14,21)(20,28)(22,23)(24,27)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=ef,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=eh,f^b=fh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #101312, #104615, #104619, … |
| maximal normal subgroup of | #101312, #104615, #104619, … |
| maximal quotient of | #244125, #244163, #244181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16793);