Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,5,7,8)(12,19)(16,21)(17,23)(22,24)(25,26,30,27)(28,33,36,31)(29,34,37,32)(35,38,40,39)(43,49)(44,52)(46,50)(47,54);(10,14)(15,20)(16,21)(22,24)(26,27)(29,37)(31,33)(35,40)(41,42,45,48)(43,50,49,46)(44,47,52,54)(51,56,55,53);(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,22,23,24)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39)(41,43,45,49)(42,46,48,50)(44,51,52,55)(47,53,54,56);(1,2)(3,5)(4,6)(7,8)(9,11,13,18)(10,15,14,20)(12,17,19,23)(16,22,21,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39)(41,42)(43,46)(44,47)(45,48)(49,50)(51,53)(52,54)(55,56);(1,3,4,7)(2,5,6,8)(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,23)(20,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)(41,44)(42,47)(43,51)(45,52)(46,53)(48,54)(49,55)(50,56)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=gh,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4462108);