Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1414 |
| automorphism | Aut(G) = #72284844 (order 2048) · Inn(G) = #354 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #354 |
| commutator | C42 | C2×C8 |
| Frattini, cosocle | #382 | C22 |
| prime radical | C23 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #382 ◁ G | |
| LECS | G ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
79 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,19,21,32)(12,23,24,39)(13,26,27,17)(14,18)(15,20,28,35)(16,25)(22,34,38,33)(29,36,30,37)(31,40);(9,11,13,21,14,22,27,38)(10,15,17,29,18,30,26,28)(12,20,25,36,24,37,40,35)(16,19,23,33,31,34,39,32)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=g,d^2=f,e^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,d^b=degh,d^c=dgh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94560, #94563, #94579, … |
| maximal normal subgroup of | #94560, #94563, #94579, … |
| maximal quotient of | #93719, #93736, #93945, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 354);