Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,22)(21,24)(25,26)(27,34)(28,32)(29,37)(30,31)(33,38)(35,36)(39,40);(1,2,5,3)(4,7,8,6)(9,11)(10,15)(12,13)(14,16)(19,24)(21,23)(26,31)(27,32)(29,38)(36,40);(2,3)(6,7)(10,12)(11,16)(17,18,20,22)(19,21,24,23)(25,27)(26,28)(29,39)(30,32)(31,34)(33,40)(35,38)(36,37);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,13,16,15)(17,18)(19,21)(20,22)(23,24)(25,28)(26,32)(27,31)(29,40)(30,34)(33,35)(36,38)(37,39);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,16)(14,15)(17,19)(18,21)(20,24)(22,23)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=fhi,c^2=gh,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4105732);