Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | Aut(G) = #72284457 (order 2048) · Inn(G) = #354 · Out(G) = #120 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #354 |
| commutator | C42 | C2×C8 |
| Frattini, cosocle | #384 | C22 |
| prime radical | C23 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ G | |
| LECS | G ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10)(11,18,20,24)(12,16,22,15)(13,17)(14,19,23,21);(9,11,13,19)(10,14)(12,21,22,20)(15,18)(16,23)(17,24)Polycyclic presentation:
2^8|a^2=d,b^2=g,c^2=gh,d^2=f,e^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=cg,d^b=deg,e^b=eh,d^c=dgh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94544, #94545, #94600, … |
| maximal normal subgroup of | #94544, #94545, #94600, … |
| maximal quotient of | #93697, #93715, #93911, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 346);