Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1323 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)(3,6)(4,5)(7,8)(25,26,31,35,30,36,29,44)(27,37,39,49,40,32,45,50)(28,34,42,53,43,52,46,33)(38,54,56,47,41,48,55,51);(2,5)(3,7)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,27,30,40)(26,32,36,37)(28,41,43,38)(29,45,31,39)(33,51,53,47)(34,54,52,48)(35,50,44,49)(42,55,46,56);(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,28,30,43)(26,33,36,53)(27,38,40,41)(29,42,31,46)(32,47,37,51)(34,35,52,44)(39,55,45,56)(48,50,54,49);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(16,20)(19,24)(25,28)(26,34)(27,38)(29,46)(30,43)(31,42)(32,48)(33,44)(35,53)(36,52)(37,54)(39,56)(40,41)(45,55)(47,49)(50,51);(2,5)(6,8)(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(25,29,30,31)(26,35,36,44)(27,39,40,45)(28,42,43,46)(32,49,37,50)(33,52,53,34)(38,55,41,56)(47,48,51,54)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfhi,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, 7366938);