Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,36,31,35,29,34)(27,32,38,46,39,47,43,49)(28,40,41,48,42,50,44,33)(37,51,53,56,54,55,52,45);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,31,39)(26,32,35,47)(28,37,42,54)(29,43,30,38)(33,45,48,56)(34,49,36,46)(40,51,50,55)(41,53,44,52);(2,5)(6,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,33)(27,37)(29,41)(30,44)(31,42)(32,45)(34,40)(35,48)(36,50)(38,52)(39,54)(43,53)(46,55)(47,56)(49,51);(10,16)(14,17)(15,22)(21,23)(25,29,31,30)(26,34,35,36)(27,38,39,43)(28,41,42,44)(32,46,47,49)(33,40,48,50)(37,52,54,53)(45,55,56,51);(1,3,4,8)(2,6,5,7)(11,19)(14,17)(18,24)(21,23)(26,35)(32,47)(33,48)(34,36)(40,50)(45,56)(46,49)(51,55)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=f,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6709070);