Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(6,8)(9,10,13,12)(11,14,15,16)(20,24)(22,23)(25,26)(27,32)(28,36)(29,34)(30,31)(33,37)(35,38)(39,40);(3,7)(6,8)(11,15)(14,16)(17,18,21,19)(20,22,24,23)(26,31)(28,37)(32,38)(34,40);(1,2,5,4)(3,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,18)(20,23,24,22)(25,27)(26,32)(28,34)(29,36)(30,35)(31,38)(33,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,16,15,14)(17,20)(18,22)(19,23)(21,24)(25,28)(26,33)(27,34)(29,38)(30,37)(31,36)(32,39)(35,40);(1,4,5,2)(3,8,7,6)(25,29,30,39)(26,34,31,40)(27,33,35,36)(28,32,37,38)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=fgi,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3348095);