Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(10,13)(11,15)(18,20)(22,24)(25,26,30,31)(27,29,35,32)(28,33,36,39)(34,38,37,40);(1,2,5,4)(3,6,8,7)(10,13)(14,16)(19,23)(22,24)(26,31)(28,36)(29,32)(34,37);(9,10)(11,14)(12,13)(15,16)(17,18,21,20)(19,22,23,24)(25,27)(26,32)(28,37)(29,31)(30,35)(33,38)(34,36)(39,40);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,23)(18,22,20,24)(25,28)(26,33)(27,34)(29,38)(30,36)(31,39)(32,40)(35,37);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,24,23,22)(25,29,30,32)(26,27,31,35)(28,38,36,40)(33,34,39,37)Polycyclic presentation:
2^9|a^2=fhi,b^2=hi,c^2=fgh,d^2=fgi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1924791);