Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | |
1730 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)(6,9)(10,12)(13,14,18,16)(15,21,23,19)(17,20,26,24)(22,28,25,27)(29,30,34,32)(31,36,39,35)(33,40,42,37)(38,43,41,44);(14,16)(19,21)(20,24)(27,28)(29,31,34,39)(30,35,32,36)(33,38,42,41)(37,43,40,44);(1,2)(3,4)(13,15,18,23)(14,19,16,21)(17,25,26,22)(20,28,24,27)(29,31)(30,36)(32,35)(33,41)(34,39)(37,43)(38,42)(40,44);(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,21,23,19)(17,24,26,20)(22,28,25,27)(29,32,34,30)(31,35,39,36)(33,40,42,37)(38,43,41,44);(5,7,8,11)(6,10,9,12)(13,17,18,26)(14,20,16,24)(15,22,23,25)(19,27,21,28)(29,33,34,42)(30,37,32,40)(31,38,39,41)(35,43,36,44)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fh,d^2=f,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4310491);