Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(8,12)(10,11)(14,18)(16,24)(19,21)(22,28)(29,30,33,34)(31,35,38,42)(32,39,41,36)(37,44,40,43);(5,6,9,7)(8,10,12,11)(14,18)(19,21)(20,25)(26,27)(30,34)(35,42)(36,39)(43,44);(2,3)(5,7,9,6)(8,11,12,10)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,31,33,38)(30,35,34,42)(32,40,41,37)(36,44,39,43);(1,3,4,2)(5,8)(6,10)(7,11)(9,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,32)(30,36)(31,37)(33,41)(34,39)(35,43)(38,40)(42,44);(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)(29,30)(31,35)(32,36)(33,34)(37,43)(38,42)(39,41)(40,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=fi,c^2=i,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4887967);