Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(6,8)(11,12)(15,16)(17,18,21,19)(20,23,24,22)(25,26)(27,33)(28,34)(29,32)(30,31)(35,38)(36,40)(37,39);(1,2,5,4)(3,6,7,8)(10,14)(15,16)(26,31)(27,34)(29,37)(36,38);(3,7)(6,8)(9,10,13,14)(11,15,12,16)(17,19,21,18)(20,23,24,22)(25,27)(26,28)(29,35)(30,34)(31,33)(32,36)(37,40)(38,39);(1,3,5,7)(2,6,4,8)(9,11,13,12)(10,15,14,16)(17,20)(18,22)(19,23)(21,24)(25,28)(26,27)(29,38)(30,33)(31,34)(32,40)(35,39)(36,37);(1,4,5,2)(3,6,7,8)(9,12,13,11)(10,16,14,15)(25,29,30,37)(26,32,31,39)(27,35,34,40)(28,36,33,38)Polycyclic presentation:
2^9|a^2=hi,b^2=fg,c^2=gi,d^2=fh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5648509);