Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309080 |
| 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 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1283 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)(3,6,7,8)(10,14)(11,17)(12,20)(22,24)(25,26,29,35,30,36,31,37)(27,38,45,49,41,32,40,54)(28,34,46,33,44,53,43,51)(39,52,55,50,42,48,56,47);(2,5)(3,7)(9,10,13,14)(11,18,17,15)(12,21,20,16)(19,22,23,24)(25,27,30,41)(26,32,36,38)(28,42,44,39)(29,45,31,40)(33,50,51,47)(34,52,53,48)(35,54,37,49)(43,55,46,56);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28,30,44)(26,33,36,51)(27,39,41,42)(29,43,31,46)(32,47,38,50)(34,37,53,35)(40,55,45,56)(48,49,52,54);(1,3,4,7)(2,6,5,8)(11,17)(15,18)(19,23)(22,24)(25,28)(26,34)(27,39)(29,46)(30,44)(31,43)(32,48)(33,35)(36,53)(37,51)(38,52)(40,56)(41,42)(45,55)(47,54)(49,50);(2,5)(6,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,29,30,31)(26,35,36,37)(27,40,41,45)(28,43,44,46)(32,49,38,54)(33,34,51,53)(39,55,42,56)(47,52,50,48)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=i,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7405762);