Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1332 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,30,34,31,35,29,33)(27,36,37,38,32,39,40,28);(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27)(26,28)(29,37)(30,40)(31,32)(33,36)(34,39)(35,38);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(25,28,31,38)(26,32,35,27)(29,39,30,36)(33,37,34,40);(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,28)(26,27)(29,39)(30,36)(31,38)(32,35)(33,40)(34,37);(1,4,5,3)(2,6,7,8)(10,18)(15,24)(16,17)(22,23)(25,29,31,30)(26,33,35,34)(27,37,32,40)(28,36,38,39)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=gh,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6397925);