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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(4,8)(11,15)(13,16)(17,18,21,19)(20,23,24,22)(25,26)(27,30)(28,33)(29,32)(31,36)(34,38)(35,40)(37,39);(2,3)(6,7)(9,10,12,14)(11,13,15,16)(17,19,21,18)(20,23,24,22)(26,27)(31,33)(32,34)(39,40);(1,2,5,3)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(25,27,30,26)(28,31,36,33)(29,34,38,32)(35,39,37,40);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,13,14,16)(17,20,21,24)(18,22,19,23)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(1,4)(2,6)(3,7)(5,8)(25,29,30,38)(26,32,27,34)(28,35,36,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=g,d^2=i,e^2=f,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, 2686905);