Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| 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 | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#386 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
854 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)(3,11,14,25)(4,15,17,8)(5,10)(6,9)(7,13,22,18)(12,27)(16,26,19,23)(20,24)(21,28)(29,32)(30,31);(1,3,5,18,6,14,20,13)(2,7,10,25,9,22,24,11)(4,12,19,28,17,29,16,30)(8,21,26,32,15,31,23,27);(1,4)(2,8)(3,12)(5,19)(6,17)(7,21)(9,15)(10,26)(11,27)(13,30)(14,29)(16,20)(18,28)(22,31)(23,24)(25,32);(2,9)(7,22)(8,15)(10,24)(11,25)(21,31)(23,26)(27,32)Polycyclic presentation:
2^8|a^2=g,b^2=eg,e^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #147803, #147841, #147848, … |
| maximal normal subgroup of | #147803, #147841, #147848, … |
| maximal quotient of | #193789, #193790, #193791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13935);