Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,8,4)(9,10,14,12)(11,16,13,15)(17,18,21,19)(20,23,24,22)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,37)(38,39);(1,3)(2,4)(5,8)(6,7)(9,10,14,12)(11,15,13,16)(17,19,21,18)(20,23,24,22)(26,31)(27,32)(33,39)(35,37);(2,6)(4,7)(9,11,14,13)(10,15,12,16)(25,27)(26,28)(29,37)(30,32)(31,34)(33,40)(35,38)(36,39);(1,4)(2,3)(5,7)(6,8)(9,12,14,10)(11,15,13,16)(17,20)(18,22)(19,23)(21,24)(25,28)(26,32)(27,31)(29,36)(30,34)(33,37)(35,39)(38,40);(1,2)(3,4)(5,6)(7,8)(9,13,14,11)(10,15,12,16)(25,29,30,38)(26,33,31,39)(27,35,32,37)(28,36,34,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=fgi,c^2=fg,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5397729);