Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1599 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)(4,7)(5,6)(9,10,14,13)(11,16,15,12)(17,18,21,20)(19,23,24,22)(25,26,29,33)(27,31,35,30)(28,32,37,40)(34,39,36,38);(2,6)(3,4)(10,13)(11,15)(18,20)(22,23)(25,27,29,35)(26,30,33,31)(28,34,37,36)(32,38,40,39);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,22)(20,23)(21,24)(25,27)(26,31)(28,36)(29,35)(30,33)(32,38)(34,37)(39,40);(1,2)(3,7)(4,8)(5,6)(9,12,14,16)(10,15,13,11)(17,18,21,20)(19,22,24,23)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,40)(35,36);(1,4,5,3)(2,8,6,7)(9,13,14,10)(11,16,15,12)(17,20,21,18)(19,23,24,22)(25,26)(27,31)(28,32)(29,33)(30,35)(34,39)(36,38)(37,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=fg,c^2=fghi,d^2=fhi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6029311);