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) = 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 | |
1708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,10)(7,11,8,12)(13,14)(15,18)(16,21)(17,20)(19,24)(22,25)(23,27)(26,28)(29,30,34,37)(31,35,32,39)(33,36,41,44)(38,42,40,43);(6,10)(11,12)(14,15)(19,21)(20,22)(26,27)(29,31,34,32)(30,35,37,39)(33,40,41,38)(36,43,44,42);(5,7,9,8)(6,11,10,12)(13,15,18,14)(16,19,24,21)(17,20,25,22)(23,27,28,26)(29,32,34,31)(30,35,37,39)(33,40,41,38)(36,42,44,43);(1,2)(3,4)(5,8,9,7)(6,11,10,12)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,30)(31,35)(32,39)(33,36)(34,37)(38,42)(40,43)(41,44);(5,6)(7,12)(8,11)(9,10)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,44)(39,43)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fgh,d^2=f,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, 5382698);