Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) = C2×#72236881 (order 2048) · Inn(G) = D42 · Out(G) = C25 |
| 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 | |
412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,12)(3,13)(4,18,20,5)(6,11,7,24)(9,23)(10,25,22,19)(14,15)(16,29)(17,28)(21,31)(26,32)(27,30);(1,3,7,16,8,17,6,23)(2,9,24,13,12,29,11,28)(4,14,25,31,20,32,19,27)(5,21,10,26,18,30,22,15);(1,4)(2,10)(3,14)(5,11)(6,19)(7,25)(8,20)(9,26)(12,22)(13,30)(15,29)(16,31)(17,32)(18,24)(21,28)(23,27);(1,5,6,22,8,18,7,10)(2,4,11,19,12,20,24,25)(3,15,16,21,17,26,23,30)(9,27,13,14,29,31,28,32)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,f^a=fh,d^b=de,e^b=eh,g^b=gh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #498172, #498173, #498367, … |
| maximal normal subgroup of | #498172, #498173, #498367, … |
| maximal quotient of | #229182, #229183, #229184, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16030);