Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #340 |
| prime radical | C23 | #106 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,4,7,5,8,12,15)(3,9,11,16,10,6,13,14)(17,18)(19,28)(20,33)(21,36)(22,38)(23,27)(24,29)(25,35)(26,34)(30,40)(31,48)(32,44)(37,41)(39,47)(42,46)(43,45);(1,3)(2,6,8,9)(4,11)(5,10)(7,14,15,16)(12,13)(17,19,21,30,34,45,35,46,23,32,20,29,22,31,37,47)(18,24,25,28,38,42,36,48,27,40,41,44,26,39,33,43)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fh,f^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,c^b=cg58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7289, #127876, #127877, … |
| maximal normal subgroup of | C2×#7289, #127876, #127877, … |
| maximal quotient of | #93406, #93430, #93456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 277);