Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12506 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
123 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18,25,32)(19,33,38,27)(20,39,34,26)(21,29,40,48)(22,28,42,46)(23,31,24,30)(35,45,41,44)(36,47,37,43);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,22,35,23,36,40,20,25,38,42,41,24,37,21,34)(18,26,29,43,30,44,46,27,32,39,48,47,31,45,28,33);(2,7)(6,14)(8,12)(9,15)(17,20,40,37,24,35,22,38,25,34,21,36,23,41,42,19)(18,27,46,45,31,43,29,39,32,33,28,44,30,47,48,26)Polycyclic presentation:
2^8|a^2=h,b^2=dgh,c^2=ef,d^2=fh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fh,g^a=gh,c^b=ch46 conjugacy classes — view the full character table.
| maximal subgroup of | #93861, #94831, #123320, … |
| maximal normal subgroup of | #93861, #94831, #123320, … |
| maximal quotient of | #135772, #141650, #141670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5502);