Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
1734 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)(6,9)(10,11)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,26)(27,28)(30,34)(31,37)(36,39)(38,43);(14,17)(19,21)(20,24)(26,27)(29,30,33,34)(31,35,37,41)(32,39,40,36)(38,44,43,42);(5,6,8,9)(7,11,12,10)(13,15)(14,19)(16,22)(17,21)(18,23)(20,26)(24,27)(25,28)(29,30)(31,35)(32,36)(33,34)(37,41)(38,42)(39,40)(43,44);(1,2)(3,4)(5,7,8,12)(6,10,9,11)(13,16,18,25)(14,20,17,24)(15,22,23,28)(19,26,21,27)(29,31)(30,35)(32,38)(33,37)(34,41)(36,42)(39,44)(40,43);(13,17,18,14)(15,21,23,19)(16,24,25,20)(22,27,28,26)(29,32,33,40)(30,36,34,39)(31,38,37,43)(35,42,41,44)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=fg,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, 4850582);