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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1710 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)(5,6,10,8)(7,12,11,9)(13,14)(15,18)(16,21)(17,20)(19,24)(22,25)(23,27)(26,28)(31,37)(35,39)(38,43)(42,44);(5,7)(6,9)(8,12)(10,11)(14,15)(19,21)(20,22)(26,27)(29,30,34,32)(31,35,37,39)(33,40,41,36)(38,44,43,42);(6,8)(9,12)(13,15,18,14)(16,19,24,21)(17,20,25,22)(23,27,28,26)(29,31)(30,35)(32,39)(33,38)(34,37)(36,42)(40,44)(41,43);(1,2)(3,4)(5,8,10,6)(7,12,11,9)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,32,34,30)(31,35,37,39)(33,40,41,36)(38,42,43,44);(5,9,10,12)(6,11,8,7)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,41)(30,36,32,40)(31,38,37,43)(35,42,39,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=h,d^2=g,e^2=gh,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, 5389086);