Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(10,14)(11,17)(16,21)(19,23)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(2,6)(3,8)(12,20)(16,21)(19,23)(22,24)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,3)(2,4)(5,8)(6,7)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,3)(2,4)(5,8)(6,7)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,27,29,35)(26,30,32,33)(28,34,36,40)(31,37,38,39);(1,4)(2,3)(5,7)(6,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)Polycyclic presentation:
2^8|a^2=fh,b^2=fh,c^2=gh,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#57918, #8101313, #8101955, … |
| maximal normal subgroup of | C2×#57918, #8101313, #8101955, … |
| maximal quotient of | #586312, #586587, #586865, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50906);