Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #466 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
303 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(17,18,22,25,23,19,21,26)(20,24,29,31,30,27,28,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32);(2,8)(7,15)(9,13)(10,16)(17,20,22,29,23,30,21,28)(18,24,25,31,19,27,26,32);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(17,21,23,22)(18,25,19,26)(20,28,30,29)(24,31,27,32)Polycyclic presentation:
2^8|a^2=f,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cgh,d^a=dh,e^a=eg,d^b=dh,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #136308, #136309, #137463, … |
| maximal normal subgroup of | #136308, #136309, #137463, … |
| maximal quotient of | #182479, #183479, #183910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10970);