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) = C26×C4 |
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 | |
1793 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)(13,14)(15,20)(16,22)(17,21)(18,19)(23,27)(24,28)(25,26)(31,37)(32,39)(34,40)(35,42);(14,19)(15,22)(17,25)(24,27)(29,30,33,36)(31,34,37,40)(32,35,39,42)(38,41,43,44);(5,6,8,9)(7,12,11,10)(13,15,18,22)(14,20,19,16)(17,23,25,28)(21,27,26,24)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16,18,20)(14,15,19,22)(17,24,25,27)(21,23,26,28)(29,31,33,37)(30,34,36,40)(32,38,39,43)(35,41,42,44);(13,17,18,25)(14,21,19,26)(15,23,22,28)(16,24,20,27)(29,32)(30,35)(31,38)(33,39)(34,41)(36,42)(37,43)(40,44)Polycyclic presentation:
2^9|b^2=fh,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, 4842510);