Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10)(11,16)(12,14)(13,15)(18,22)(20,23);(2,5)(6,8)(10,12)(15,16)(18,22)(19,24);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,12,16)(17,18)(19,23)(20,24)(21,22);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,16,13,15)(17,19,21,24)(18,23,22,20);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,16,12,15)(17,20,21,23)(18,24,22,19)Polycyclic presentation:
2^8|c^2=f,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6471908, #6471912, #6471917, … |
| maximal normal subgroup of | #6471908, #6471912, #6471917, … |
| maximal quotient of | #553816, #553838, #554060, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30322);