Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(11,17)(15,20)(18,23)(22,24)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(10,14)(11,17)(12,19)(22,24)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,3,4,7)(2,5,6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,d^2=gh,e^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#46297, #7702975, #7703001, … |
| maximal normal subgroup of | C2×#46297, #7702975, #7703001, … |
| maximal quotient of | #557187, #557199, #557512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39285);