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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(11,17)(15,20)(18,23)(22,24)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(10,14)(11,17)(12,19)(22,24)(25,26,29,32)(27,30,35,33)(28,36,37,31)(34,39,40,38);(1,3,5,8)(2,6,4,7)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(1,2,5,4)(3,6,8,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=fgh,d^2=fg,e^2=g,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×#45951, #7702451, #7702513, … |
| maximal normal subgroup of | C2×#45951, #7702451, #7702513, … |
| maximal quotient of | #557202, #557214, #557517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38939);