Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1618 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)(5,6,8,9)(7,11,12,10)(16,23)(20,25)(21,27)(26,28)(29,30)(31,34)(32,39)(33,36)(35,41)(37,44)(38,43)(40,42);(6,9)(10,11)(14,18)(19,24)(20,25)(26,28)(29,30,33,36)(31,34,38,43)(32,35,41,39)(37,42,40,44);(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,19,22,24)(16,20,23,25)(21,26,27,28)(29,30)(31,34)(32,35)(33,36)(37,42)(38,43)(39,41)(40,44);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,15,17,22)(14,19,18,24)(16,21,23,27)(20,26,25,28)(29,31,33,38)(30,34,36,43)(32,40,41,37)(35,44,39,42);(5,7,8,12)(6,10,9,11)(13,16)(14,20)(15,21)(17,23)(18,25)(19,26)(22,27)(24,28)(29,32,33,41)(30,35,36,39)(31,37,38,40)(34,42,43,44)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=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4775969);