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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1676 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)(3,7)(4,6)(5,8)(10,13)(14,15)(17,18)(19,24)(20,22)(21,23)(25,26,30,28)(27,31,34,35)(29,36,37,32)(33,39,40,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,20)(18,22,23,24)(26,28)(31,35)(32,36)(38,39);(2,6)(5,7)(9,10,12,13)(11,14,16,15)(17,18,21,23)(19,24,20,22)(25,27,30,34)(26,31,28,35)(29,33,37,40)(32,38,36,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,24)(21,23)(25,28,30,26)(27,31,34,35)(29,32,37,36)(33,39,40,38);(1,3)(2,7)(4,8)(5,6)(9,11,12,16)(10,14,13,15)(17,20,21,19)(18,22,23,24)(25,29,30,37)(26,32,28,36)(27,33,34,40)(31,38,35,39)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,c^2=fgh,d^2=fg,e^2=fgh,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, 2680870);