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 | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,4)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,26,30,31)(27,28,32,35)(29,37,38,33)(34,40,39,36);(1,2,6,5)(3,8,4,7)(10,14)(11,17)(12,21)(22,24)(26,31)(28,35)(33,37)(36,40);(1,3,6,4)(2,7,5,8)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,27,30,32)(26,28,31,35)(29,34,38,39)(33,36,37,40);(1,4,6,3)(2,8,5,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28)(26,32)(27,31)(29,36)(30,35)(33,39)(34,37)(38,40);(1,5,6,2)(3,8,4,7)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,37)(32,39)(35,40)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=f,d^2=g,e^2=g,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×#54666, #8100793, #8101190, … |
| maximal normal subgroup of | C2×#54666, #8100793, #8101190, … |
| maximal quotient of | #586054, #586394, #586624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47654);