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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
696 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)(3,6)(4,5)(7,8)(9,10,14,12)(11,15,19,17)(13,20,21,16)(18,23,24,22)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(1,3,5,8)(2,6,4,7)(10,12)(15,17)(16,20)(22,23)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(2,4)(6,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,4,5,2)(3,7,8,6)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,3)(2,6)(4,7)(5,8)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^8|a^2=fh,b^2=gh,c^2=fh,d^2=fg,e^2=f,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×#57640, #8101914, #8101949, … |
| maximal normal subgroup of | C2×#57640, #8101914, #8101949, … |
| maximal quotient of | #586044, #587054, #587132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 50628);