Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | Aut(G) = #72337551 (order 2048) · Inn(G) = D42 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2361 |
| prime radical | #3067 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12)(3,13)(4,5,22,25)(6,10,14,11)(8,18)(9,20,24,21)(15,26)(16,32)(17,28)(19,29)(23,27)(30,31);(1,3,14,17,7,19,6,18)(2,8,11,13,12,28,10,29)(4,15,21,31,22,32,20,27)(5,23,9,26,25,30,24,16);(1,4)(2,9)(3,15)(5,10)(6,20)(7,22)(8,26)(11,25)(12,24)(13,30)(14,21)(16,28)(17,31)(18,27)(19,32)(23,29);(1,5,6,24,7,25,14,9)(2,4,10,20,12,22,11,21)(3,16,17,23,19,26,18,30)(8,27,13,15,28,31,29,32)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=de,e^b=eh,g^b=gh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #498124, #498125, #498379, … |
| maximal normal subgroup of | #498124, #498125, #498379, … |
| maximal quotient of | #229185, #229186, #229187, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16031);