Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,12,14)(11,15,16,13)(18,22)(20,23)(25,26,30,31)(27,32,36,39)(28,29,37,33)(34,40,38,35);(1,2,5,4)(3,6,7,8)(9,10,12,14)(11,13,16,15)(18,22)(19,24)(26,31)(28,37)(32,39)(34,38);(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,27,30,36)(26,32,31,39)(28,38,37,34)(29,35,33,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,24)(18,23,22,20)(25,28,30,37)(26,33,31,29)(27,34,36,38)(32,35,39,40);(1,4,5,2)(3,8,7,6)(9,11)(10,13)(12,16)(14,15)(17,20,21,23)(18,24,22,19)(25,29,30,33)(26,28,31,37)(27,35,36,40)(32,38,39,34)Polycyclic presentation:
2^9|a^2=fgh,b^2=fi,c^2=f,d^2=fh,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2702040);