Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7440 |
| automorphism | Aut(G) = #72282281 (order 1024) · Inn(G) = #1353 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1353 |
| commutator | C2×M16 | C2×C4 |
| Frattini, cosocle | #423 | C22 |
| prime radical | C23 | #104 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #423 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,11,13,23,15,28,14,31)(4,8,17,24)(5,19)(6,9)(7,12,21,29,22,30,27,16)(10,20)(18,26,32,25);(1,3)(2,7,9,22)(4,16,17,29)(5,14)(6,15)(8,23)(10,21,19,27)(11,26)(12,18,30,32)(13,20)(24,31)(25,28)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fgh,d^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fh,c^b=cf,d^b=defg,g^b=gh,d^c=dg,e^c=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7441, #150309, #150451, … |
| maximal normal subgroup of | C2×#7441, #150309, #150451, … |
| maximal quotient of | #93941, #93988, #94338, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 429);