Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(10,13)(11,15)(17,18,21,20)(19,22,23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,37)(38,39);(9,10,12,13)(11,14,15,16)(19,23)(22,24)(26,31)(28,32)(34,39)(36,40);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,13)(14,15)(17,19,21,23)(18,22,20,24)(25,27,30,33)(26,32,31,28)(29,37,38,35)(34,36,39,40);(1,2)(3,6)(4,7)(5,8)(17,18,21,20)(19,22,23,24)(25,28,30,32)(26,33,31,27)(29,36,38,40)(34,37,39,35);(1,4,5,3)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,24,23,22)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,40)(33,37)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=fhi,d^2=fh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2456197);