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:
(2,4)(3,5)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(1,2)(3,7)(4,6)(5,8)(10,11)(12,19)(13,21)(22,23)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,3,6,5)(2,7,4,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,4,6,2)(3,7,5,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,5,6,3)(2,7,4,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,c^2=f,d^2=gh,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×#58535, #8104385, #8104430, … |
| maximal normal subgroup of | C2×#58535, #8104385, #8104430, … |
| maximal quotient of | #585670, #586930, #587292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51523);