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 | |
1258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,3)(4,7,8,6)(10,14)(11,17)(16,21)(19,23)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40)(41,42,46,44)(43,49,50,47)(45,52,53,48)(51,54,56,55);(1,3,5,2)(4,7,8,6)(12,20)(16,21)(19,23)(22,24)(26,27)(31,33)(32,34)(38,39)(43,50)(45,53)(47,49)(48,52);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,27,30,26)(28,31,36,33)(29,32,37,34)(35,39,40,38)(41,43)(42,47)(44,49)(45,51)(46,50)(48,54)(52,55)(53,56);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39)(41,44,46,42)(43,47,50,49)(45,52,53,48)(51,54,56,55);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)(41,45)(42,48)(43,51)(44,52)(46,53)(47,54)(49,55)(50,56)Polycyclic presentation:
2^9|a^2=gi,b^2=fghi,c^2=f,d^2=h,e^2=f,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, 5394938);