Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1880 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)(6,9)(7,11)(15,21)(16,23)(19,24)(20,25)(29,30,35,40)(31,37,32,36)(33,38,39,34)(41,44,43,42);(7,11)(10,12)(14,18)(19,24)(20,25)(26,28)(29,31,35,32)(30,36,40,37)(33,43,39,41)(34,42,38,44);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,32,35,31)(30,37,40,36)(33,43,39,41)(34,42,38,44);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,33,35,39)(30,38,40,34)(31,41,32,43)(36,42,37,44);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,34,35,38)(30,39,40,33)(31,42,32,44)(36,41,37,43)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=f,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4507323);