Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1329 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,15,12)(11,18,20,16)(13,21,14,17)(19,23,22,24)(25,26)(27,36)(28,33)(29,34)(30,43)(31,35)(32,38)(37,51)(39,46)(40,48)(41,50)(42,49)(44,47)(45,53)(52,55)(54,56);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,18)(13,19)(14,22)(15,20)(17,23)(21,24);(3,8)(5,7)(9,12,15,10)(11,18,20,16)(13,17,14,21)(19,23,22,24)(25,27)(26,32)(28,37)(29,38)(30,44)(31,39)(33,45)(34,46)(35,47)(36,43)(40,52)(41,54)(42,53)(48,51)(49,56)(50,55);(2,6)(5,7)(9,13,15,14)(10,17,12,21)(11,19,20,22)(16,23,18,24)(25,28,31,41)(26,33,35,50)(27,37,39,54)(29,42,30,40)(32,45,47,55)(34,49,43,48)(36,51,46,56)(38,53,44,52);(9,14,15,13)(10,17,12,21)(11,19,20,22)(16,24,18,23)(25,29,31,30)(26,34,35,43)(27,38,39,44)(28,40,41,42)(32,46,47,36)(33,48,50,49)(37,52,54,53)(45,51,55,56)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fh,d^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,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, 6901010);