Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#106 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | C23 | #106 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,20,23,35)(12,17)(13,26)(14,29)(15,19)(16,22,31,38)(18,27)(21,34,24,36)(25,33)(28,40)(30,37,32,39);(9,11,13,22,14,21,27,37,15,23,28,38,12,24,25,39)(10,16,18,20,17,32,26,36,19,31,33,35,29,30,40,34);(2,5)(6,8)(9,12,15,14)(10,17,19,29)(11,21,23,24)(13,25,28,27)(16,30,31,32)(18,26,33,40)(20,34,35,36)(22,37,38,39)Polycyclic presentation:
2^8|a^2=e,b^2=d,c^2=h,d^2=f,e^2=g,f^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94407, #94418, #135075, … |
| maximal normal subgroup of | #94407, #94418, #135075, … |
| maximal quotient of | #126692, #129105, #134658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5009);