Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10,14,15)(11,17,19,12)(13,21,22,16)(18,20,24,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38);(1,2,4,6)(3,5,7,8)(10,15)(12,17)(16,21)(20,23)(26,29)(27,33)(32,37)(34,36);(1,3,4,7)(2,5,6,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,23)(22,24)(25,27,30,33)(26,31,29,35)(28,36,38,34)(32,40,37,39);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,24)(22,23)(25,28,30,38)(26,32,29,37)(27,34,33,36)(31,39,35,40);(9,13,14,22)(10,16,15,21)(11,18,19,24)(12,20,17,23)(25,29,30,26)(27,35,33,31)(28,37,38,32)(34,40,36,39)Polycyclic presentation:
2^8|a^2=fh,b^2=fgh,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #132489, #253529, #253562, … |
| maximal normal subgroup of | #132489, #253529, #253562, … |
| maximal quotient of | #560389, #560526, #561574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44219);