Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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 | |
1720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,10,9)(7,12,11,8)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,26)(27,28)(29,30,33,36)(31,37,38,34)(32,40,41,35)(39,42,44,43);(5,6,10,9)(7,8,11,12)(14,18)(16,25)(19,21)(22,28)(31,38)(34,37)(39,44)(42,43);(5,7)(6,8)(9,12)(10,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,31,33,38)(30,34,36,37)(32,39,41,44)(35,42,40,43);(1,2)(3,4)(5,8,10,12)(6,11,9,7)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27)(29,30)(31,34)(32,35)(33,36)(37,38)(39,42)(40,41)(43,44);(5,9,10,6)(7,12,11,8)(13,16,17,25)(14,20,18,24)(15,22,23,28)(19,26,21,27)(29,32,33,41)(30,35,36,40)(31,39,38,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=fh,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4223172);