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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1788 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)(5,6,10,12)(7,8,9,11)(13,14)(15,18)(16,19)(17,20)(21,24)(22,25)(23,26)(27,28)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(14,15)(17,25)(19,21)(23,28)(29,30,33,36)(31,34,39,37)(32,35,41,40)(38,42,44,43);(5,7,10,9)(6,11,12,8)(13,15,18,14)(16,21,24,19)(17,22,25,20)(23,27,28,26)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(1,2)(3,4)(5,8,10,11)(6,7,12,9)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,9,10,7)(6,11,12,8)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=fgh,d^2=g,e^2=fg,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, 4849885);