Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | |
1305 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,31)(28,32)(29,34)(30,33)(35,44)(36,46)(37,47)(38,48)(39,49)(40,50)(41,45)(42,52)(43,51)(53,55)(54,56);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,30,37)(26,31,33,47)(28,38,39,54)(29,41,43,36)(32,48,49,56)(34,45,51,46)(35,42,53,40)(44,52,55,50);(2,5)(6,8)(25,28,30,39)(26,32,33,49)(27,35,37,53)(29,42,43,40)(31,44,47,55)(34,52,51,50)(36,54,41,38)(45,48,46,56);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(28,39)(32,49)(35,53)(38,54)(40,42)(44,55)(48,56)(50,52);(1,3,4,7)(2,6,5,8)(11,19)(14,17)(18,24)(22,23)(26,33)(27,36)(29,43)(31,45)(32,49)(35,54)(37,41)(38,53)(40,42)(44,48)(46,47)(55,56)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6619894);