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 | |
1669 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)(7,11)(10,12)(13,14)(15,20)(16,21)(17,22)(18,19)(23,28)(24,26)(25,27)(30,34)(31,37)(36,40)(39,43);(6,9)(10,12)(14,19)(15,21)(17,25)(24,28)(29,30,33,34)(31,35,37,38)(32,36,41,40)(39,42,43,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,21)(14,20,19,16)(17,26,25,23)(22,24,27,28)(29,30)(31,38)(32,40)(33,34)(35,37)(36,41)(39,42)(43,44);(1,2)(3,4)(13,16,18,20)(14,21,19,15)(17,24,25,28)(22,26,27,23)(29,31)(30,35)(32,39)(33,37)(34,38)(36,42)(40,44)(41,43);(5,7,8,11)(6,10,9,12)(13,17,18,25)(14,22,19,27)(15,23,21,26)(16,24,20,28)(29,32,33,41)(30,36,34,40)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|b^2=g,c^2=fg,d^2=fg,e^2=h,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, 5385639);