Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #357 |
| commutator | C16 | C42 |
| Frattini, cosocle | C22×C16 | C22 |
| socle | C2 | #1380 |
| prime radical | C23 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
83 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,8,15,29,14)(3,16,18,48,21,40,52,50)(4,22,25,46,27,45,63,12)(6,30,28,42,32,41,26,10)(7,33,24,13,31,43,62,44)(9,34,35,23,39,60,38,17)(19,54,53,47,56,37,61,64)(20,57,51,49,55,36,59,58);(1,3,6,19,31,55,27,17,29,52,26,61,24,51,25,60,8,21,32,56,7,20,4,23,5,18,28,53,62,59,63,34)(2,9,12,36,44,54,42,16,11,35,22,58,33,47,41,48,15,39,46,57,13,37,10,40,14,38,45,49,43,64,30,50)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=fh,f^2=gh,g^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fg,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7480, #151968, #151971, … |
| maximal normal subgroup of | C2×#7480, #151968, #151971, … |
| maximal quotient of | #94026, #94048, #94050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 468);