Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1199 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)(3,7)(4,8)(5,6)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,26,30,34)(27,31,38,45)(28,32,39,46)(29,33,41,48)(35,42,37,44)(36,43,51,55)(40,47,52,56)(49,53,50,54);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,30,38)(26,31,34,45)(28,36,39,51)(29,35,41,37)(32,43,46,55)(33,42,48,44)(40,49,52,50)(47,53,56,54);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(27,35)(28,39)(29,41)(31,42)(32,46)(33,48)(36,50)(37,38)(43,54)(44,45)(49,51)(53,55);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(25,28,30,39)(26,32,34,46)(27,36,38,51)(29,40,41,52)(31,43,45,55)(33,47,48,56)(35,49,37,50)(42,53,44,54);(1,4,5,3)(2,8,6,7)(11,19)(14,17)(18,24)(21,23)(27,35)(29,41)(31,42)(33,48)(36,49)(37,38)(40,52)(43,53)(44,45)(47,56)(50,51)(54,55)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=f,d^2=gi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6639994);