Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1632 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)(5,6,9,10)(7,8,12,11)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,25)(26,28)(30,34)(32,39)(35,41)(37,40);(6,10)(7,12)(13,14,17,20)(15,18,23,21)(16,19,25,24)(22,26,28,27)(29,30)(31,35)(32,36)(33,34)(37,43)(38,41)(39,42)(40,44);(6,10)(8,11)(13,14)(15,18)(16,24)(17,20)(19,25)(21,23)(22,27)(26,28)(29,31,33,38)(30,35,34,41)(32,40,39,37)(36,44,42,43);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,15,17,23)(14,18,20,21)(16,22,25,28)(19,26,24,27)(29,30)(31,35)(32,36)(33,34)(37,43)(38,41)(39,42)(40,44);(5,8,9,11)(6,7,10,12)(13,16,17,25)(14,19,20,24)(15,22,23,28)(18,26,21,27)(29,32,33,39)(30,36,34,42)(31,37,38,40)(35,43,41,44)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fgh,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4508963);