Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632088 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,29,34)(27,36,38,31)(28,40,42,33)(30,32,44,35)(37,46,41,49)(39,47,52,45)(43,50,54,48)(51,55,53,56);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27,29,38)(26,31,34,36)(28,41,42,37)(30,39,44,52)(32,45,35,47)(33,49,40,46)(43,53,54,51)(48,56,50,55);(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(26,32)(30,44)(31,45)(33,48)(34,35)(36,47)(39,52)(40,50)(43,54)(46,55)(49,56)(51,53);(2,5)(6,7)(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,28,29,42)(26,33,34,40)(27,37,38,41)(30,43,44,54)(31,46,36,49)(32,48,35,50)(39,51,52,53)(45,55,47,56);(1,3,4,8)(2,6,5,7)(10,18)(15,24)(16,17)(22,23)(26,34)(27,38)(28,42)(32,35)(39,52)(43,54)(46,49)(55,56)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=h,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6559547);