Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
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 | |
451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,25)(15,22,27,28)(16,20,26,21)(17,24,18,23);(6,9)(10,12)(13,15,18,26,19,27,17,16)(14,20,24,28,25,21,23,22);(1,2,3,4)(13,15,17,16,19,27,18,26)(14,21,23,28,25,20,24,22);(5,7,8,11)(6,10,9,12)(13,16,17,27,19,26,18,15)(14,22,23,21,25,28,24,20)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=fh,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=gh,c^b=ch,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #125421, #125538, #294768, … |
| maximal normal subgroup of | #125421, #125538, #294768, … |
| maximal quotient of | #352681, #352687, #352689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26243);