Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10208 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11)(4,7)(5,9)(6,10)(8,12)(13,15)(14,16)(17,18)(19,26)(20,30)(21,24)(22,25)(23,27)(28,31)(29,32);(2,7)(4,5)(8,11)(9,10)(12,14)(15,16)(17,19)(18,23)(20,29)(21,27)(22,28)(24,31)(25,32)(26,30);(1,3)(2,8)(4,12)(5,14)(6,13)(7,15)(9,11)(10,16)(19,27)(20,21)(23,31)(24,30)(26,32)(28,29)Polycyclic presentation:
2^8|d^2=g,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fh,d^c=dh,e^c=egh,f^c=fh25 conjugacy classes — view the full character table.
| Fitting subgroup of | #10208⋊C3 |
| maximal subgroup of | C2×#10208, #232817, #232820, … |
| maximal normal subgroup of | C2×#10208, #232817, #232820, … |
| maximal quotient of | #102685, #102689, #102703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3196);