Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
125 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10,15,27,26,37,25,17,14,20,12,18,13,19,29,36)(11,16,28,30,23,32,38,35,24,34,22,31,21,33,39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,22)(13,21)(14,24)(15,28)(17,35)(18,31)(19,33)(20,34)(23,26)(25,38)(27,30)(29,39)(32,37)(36,40);(1,4,5,3)(2,6,7,8)(10,17)(11,21)(12,25)(13,26)(15,29)(16,30)(18,19)(20,36)(22,28)(23,24)(27,37)(31,34)(32,40)(33,35)Polycyclic presentation:
2^8|a^2=eh,b^2=d,c^2=d,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,c^b=cf,f^b=fh,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93822, #93823, C2×#13518, … |
| maximal normal subgroup of | #93822, #93823, C2×#13518, … |
| maximal quotient of | #146145, #146156, #146172, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6506);