Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,16)(12,14)(13,15)(17,18,21,24)(19,20,23,22)(25,26)(27,35)(28,29)(30,31)(32,38)(33,34)(36,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,11,13,14)(10,12,15,16)(17,19)(18,22)(20,24)(21,23)(26,31)(28,34)(32,35)(36,40);(17,18,21,24)(19,20,23,22)(25,27,30,38)(26,32,31,35)(28,36,34,40)(29,37,33,39);(1,4,5,6)(2,3,7,8)(9,10)(11,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,28)(26,33)(27,36)(29,31)(30,34)(32,39)(35,37)(38,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,14)(11,15)(13,16)(17,20)(18,23)(19,24)(21,22)(25,29,30,33)(26,34,31,28)(27,37,38,39)(32,40,35,36)Polycyclic presentation:
2^9|a^2=ghi,b^2=fhi,c^2=gh,d^2=gi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2566683);