Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(10,13)(14,16)(19,20)(23,24)(25,26,30,34)(27,28,32,31)(29,37,39,33)(35,40,38,36);(18,22)(23,24)(25,27,30,32)(26,31,34,28)(29,38,39,35)(33,36,37,40);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,18,21,22)(19,23,20,24)(25,27)(26,31)(28,34)(29,35)(30,32)(33,40)(36,37)(38,39);(1,2)(3,6)(4,5)(7,8)(17,19,21,20)(18,23,22,24)(25,28,30,31)(26,32,34,27)(29,36,39,40)(33,38,37,35);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,24,22,23)(25,29)(26,33)(27,35)(28,36)(30,39)(31,40)(32,38)(34,37)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgi,d^2=h,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4240757);