Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(6,9)(10,11)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26)(30,31)(33,41)(35,37)(39,44);(14,18)(16,24)(19,21)(22,28)(29,30,34,31)(32,35,40,37)(33,38,41,36)(39,43,44,42);(5,6)(7,11)(8,9)(10,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28)(29,31,34,30)(32,37,40,35)(33,38,41,36)(39,43,44,42);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,32,34,40)(30,35,31,37)(33,39,41,44)(36,42,38,43);(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)(29,33,34,41)(30,36,31,38)(32,39,40,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=fh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4851502);