Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | Aut(G) = #72337551 (order 2048) · Inn(G) = C2×#121 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C4×#121 |
| prime radical | #3067 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,13,7,11)(3,14,5,9,17,24,16,25)(4,18,20,10,21,26,22,27)(15,29,19,28,30,23,31,32);(1,3,6,5,8,17,7,16)(2,9,11,14,13,25,12,24)(4,15,20,19,21,30,22,31)(10,23,18,28,27,29,26,32);(1,4)(2,10)(3,15)(5,19)(6,20)(7,22)(8,21)(9,23)(11,18)(12,26)(13,27)(14,28)(16,31)(17,30)(24,32)(25,29);(1,5,6,17,8,16,7,3)(2,9,12,24,13,25,11,14)(4,19,20,30,21,31,22,15)(10,23,26,32,27,29,18,28)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=g,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #495739, #495740, #495899, … |
| maximal normal subgroup of | #495739, #495740, #495899, … |
| maximal quotient of | #211042, #211043, #211044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14547);