Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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×#385 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,10,5,21)(3,12,14,8,16,24,15,23)(4,17,19,9,20,26,18,25)(13,28,29,27,30,22,31,32);(1,3,6,14,7,16,5,15)(2,8,11,24,10,23,21,12)(4,13,19,29,20,30,18,31)(9,22,26,32,25,28,17,27);(1,4)(2,9)(3,13)(5,18)(6,19)(7,20)(8,22)(10,25)(11,26)(12,27)(14,29)(15,31)(16,30)(17,21)(23,28)(24,32);(2,10)(8,23)(9,25)(11,21)(12,24)(17,26)(22,28)(27,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,e^b=eh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #365173, #365449, #365457, … |
| maximal normal subgroup of | #365173, #365449, #365457, … |
| maximal quotient of | #191809, #191810, #191814, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13855);