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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,12,13)(11,15,16,14)(17,18)(19,22)(20,24)(21,23)(26,27)(28,35)(29,37)(39,40);(11,16)(14,15)(17,19)(18,22)(20,23)(21,24)(25,26,30,27)(28,31,35,33)(29,34,37,32)(36,40,38,39);(10,13)(14,15)(18,23)(19,24)(25,27,30,26)(28,31,35,33)(29,32,37,34)(36,40,38,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,24)(22,23)(25,28,30,35)(26,31,27,33)(29,38,37,36)(32,40,34,39);(1,3,4,7)(2,5,6,8)(9,11,12,16)(10,14,13,15)(17,20)(18,19)(21,22)(23,24)(25,29,30,37)(26,32,27,34)(28,36,35,38)(31,39,33,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,c^2=f,d^2=f,e^2=hi,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, 4185604);