Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5020720 |
| 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 | |
1766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6,8,9)(7,10,11,12)(14,18)(16,24)(19,21)(22,28)(29,30,35,38)(31,32,33,36)(34,37,43,44)(39,40,41,42);(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28)(29,31,35,33)(30,36,38,32)(34,39,43,41)(37,42,44,40);(6,9)(10,12)(13,14)(15,21)(16,25)(17,18)(19,23)(20,24)(22,26)(27,28)(29,32,35,36)(30,31,38,33)(34,42,43,40)(37,41,44,39);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,33,35,31)(30,32,38,36)(34,41,43,39)(37,40,44,42);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)(29,34)(30,37)(31,39)(32,40)(33,41)(35,43)(36,42)(38,44)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4928648);