Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1331 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26,30,35)(27,31,38,46)(28,39,40,33)(29,32,42,34)(36,49,50,44)(37,43,52,45)(41,48,53,47)(51,55,56,54);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,30,38)(26,31,35,46)(28,36,40,50)(29,37,42,52)(32,43,34,45)(33,44,39,49)(41,51,53,56)(47,54,48,55);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(29,42)(31,43)(33,47)(34,35)(37,52)(39,48)(41,53)(44,54)(45,46)(49,55)(51,56);(2,6)(3,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,30,40)(26,33,35,39)(27,36,38,50)(29,41,42,53)(31,44,46,49)(32,47,34,48)(37,51,52,56)(43,54,45,55);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(28,40)(33,39)(36,50)(41,53)(44,49)(47,48)(51,56)(54,55)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7212010);