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) = C26 |
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 | |
1540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10)(11,12)(13,16)(14,15)(17,18,21,20)(19,22,23,24)(25,26,30,28)(27,32,35,31)(29,33,38,36)(34,40,37,39);(9,11)(10,14)(12,16)(13,15)(19,23)(22,24)(25,27,30,35)(26,31,28,32)(29,34,38,37)(33,39,36,40);(9,12,13,14)(10,15,16,11)(17,19)(18,22)(20,24)(21,23)(25,27)(26,32)(28,31)(29,37)(30,35)(33,39)(34,38)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(25,28,30,26)(27,31,35,32)(29,33,38,36)(34,40,37,39);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,16)(13,14)(17,20,21,18)(19,24,23,22)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=g,d^2=fg,e^2=fhi,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, 3520995);