Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #391 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1446 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,7,12,21)(4,14)(5,17)(6,9)(8,16)(10,18)(11,24,26,29)(13,28,27,20)(15,30)(19,32,22,25)(23,31);(1,3)(2,7,9,21)(4,11)(5,13)(6,12)(8,22,23,19)(10,20,17,28)(14,29,30,24)(15,26)(16,25)(18,27)(31,32);(1,4)(2,8,9,23)(3,11)(5,16)(6,15)(7,19,21,22)(10,14,17,30)(12,26)(13,25)(18,31)(20,24,28,29)(27,32)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=dh,g^b=gh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11389, #358818, #362913, … |
| maximal normal subgroup of | C2×#11389, #358818, #362913, … |
| maximal quotient of | #121999, #122000, #122001, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4377);