Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18,22,26)(19,24,29,32)(20,30,28,23)(21,31,27,25);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,27,20,22,29,21,28)(18,23,31,32,26,30,25,24);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,21,19,22,28,27,29)(18,24,25,30,26,32,31,23);(2,7)(6,8)(17,20,27,29,22,28,21,19)(18,24,31,23,26,32,25,30)Polycyclic presentation:
2^8|a^2=gh,b^2=eh,c^2=e,d^2=fh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28061, #6457506, #6457512, … |
| maximal normal subgroup of | C2×#28061, #6457506, #6457512, … |
| maximal quotient of | #361100, #362404, #367761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21049);