Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(11,17)(15,20)(18,23)(22,24)(25,26,31,29)(27,34,28,32)(30,37,38,33)(35,39,36,40);(10,14)(11,17)(12,19)(22,24)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(1,3,4,8)(2,5,6,7)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,29,31,26)(27,32,28,34)(30,37,38,33)(35,39,36,40);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,30,31,38)(26,33,29,37)(27,35,28,36)(32,39,34,40)Polycyclic presentation:
2^8|a^2=fg,b^2=f,c^2=fgh,d^2=fh,e^2=f,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 | #97370, #266577, #266581, … |
| maximal normal subgroup of | #97370, #266577, #266581, … |
| maximal quotient of | #557116, #557403, #557802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39231);