Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#120 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#386 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,18,23)(3,7,13,21,12,20,27,31)(4,14,16,24,15,8,17,22)(11,19,26,30,25,29,28,32);(1,3,5,13,6,12,18,27)(2,7,10,21,9,20,23,31)(4,11,17,28,15,25,16,26)(8,19,24,32,14,29,22,30);(1,4)(2,8)(3,11)(5,17)(6,15)(7,19)(9,14)(10,24)(12,25)(13,28)(16,18)(20,29)(21,32)(22,23)(26,27)(30,31);(2,9)(7,20)(8,14)(10,23)(19,29)(21,31)(22,24)(30,32)Polycyclic presentation:
2^8|a^2=g,b^2=eg,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,f^b=fh,e^c=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #147802, #147841, #147844, … |
| maximal normal subgroup of | #147802, #147841, #147844, … |
| maximal quotient of | #193865, #193866, #193867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13952);