Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14,17,20)(15,21,22,18)(16,19,24,26)(23,27,28,25)(29,30,35,32)(31,38,33,36)(34,40,42,37)(39,43,41,44);(6,9)(10,12)(15,22)(18,21)(23,28)(25,27)(29,31,35,33)(30,36,32,38)(34,39,42,41)(37,43,40,44);(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28)(29,32,35,30)(31,36,33,38)(34,40,42,37)(39,43,41,44);(1,2)(3,4)(13,15)(14,18)(16,23)(17,22)(19,25)(20,21)(24,28)(26,27)(29,33,35,31)(30,36,32,38)(34,39,42,41)(37,44,40,43);(5,7)(6,10)(8,11)(9,12)(13,16,17,24)(14,19,20,26)(15,23,22,28)(18,25,21,27)(29,34)(30,37)(31,39)(32,40)(33,41)(35,42)(36,43)(38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=h,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4769991);