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 | |
1490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,23,20,22)(25,26,29,30)(27,33,35,31)(28,32,37,38)(34,40,36,39);(1,2,5,4)(3,6,7,8)(11,16)(13,15)(17,18,21,24)(19,22,20,23)(26,30)(31,33)(32,38)(39,40);(9,11)(10,13)(12,16)(14,15)(17,19,21,20)(18,22,24,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,27,29,35)(26,31,30,33)(28,36,37,34)(32,40,38,39);(1,4,5,2)(3,8,7,6)(9,10)(11,13)(12,14)(15,16)(17,20,21,19)(18,23,24,22)(25,28,29,37)(26,32,30,38)(27,34,35,36)(31,39,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=gh,d^2=f,e^2=fhi,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, 2707809);