Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(3,7)(5,8)(10,14)(11,17)(16,21)(19,23)(26,30)(27,33)(28,36)(38,40)(41,42,46,43)(44,47,52,49)(45,48,53,50)(51,54,56,55);(1,2,4,6)(3,5,7,8)(12,20)(16,21)(19,23)(22,24)(26,30)(28,36)(31,34)(35,39)(42,43)(45,53)(47,49)(51,56);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,37)(38,39)(41,43,46,42)(44,49,52,47)(45,48,53,50)(51,54,56,55);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,27,29,33)(26,31,30,34)(28,35,36,39)(32,38,37,40)(41,44,46,52)(42,47,43,49)(45,51,53,56)(48,54,50,55);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,28,29,36)(26,32,30,37)(27,35,33,39)(31,38,34,40)(41,45)(42,48)(43,50)(44,51)(46,53)(47,54)(49,55)(52,56)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=f,d^2=ghi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5428988);