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 | |
1820 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)(7,11)(10,12)(16,23)(20,25)(21,27)(26,28)(29,30,33,34)(31,37,39,35)(32,36,40,41)(38,43,44,42);(5,6,9,8)(7,10,11,12)(14,18)(19,24)(20,25)(26,28)(30,34)(32,40)(35,37)(38,44);(5,7)(6,10)(8,12)(9,11)(13,14,17,18)(15,19,22,24)(16,20,23,25)(21,26,27,28)(29,30)(31,35)(32,36)(33,34)(37,39)(38,42)(40,41)(43,44);(1,2)(3,4)(5,8,9,6)(7,10,11,12)(13,15,17,22)(14,19,18,24)(16,21,23,27)(20,26,25,28)(29,31)(30,35)(32,38)(33,39)(34,37)(36,42)(40,44)(41,43);(13,16)(14,20)(15,21)(17,23)(18,25)(19,26)(22,27)(24,28)(29,32,33,40)(30,36,34,41)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=i,b^2=fg,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, 4850959);