Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1162 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)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,30,34)(27,31,37,44)(28,32,40,47)(29,33,41,48)(35,42,50,54)(36,43,39,46)(38,45,51,55)(49,53,52,56);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(27,35)(28,38)(29,41)(31,42)(32,45)(33,48)(36,39)(37,50)(40,51)(43,46)(44,54)(47,55);(2,6)(7,8)(25,27,29,35,30,37,41,50)(26,31,33,42,34,44,48,54)(28,39,38,52,40,36,51,49)(32,46,45,56,47,43,55,53);(1,3,5,4)(2,7,6,8)(10,15)(14,17)(16,23)(22,24)(28,40)(32,47)(36,39)(38,51)(43,46)(45,55)(49,52)(53,56);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,28,41,51,30,40,29,38)(26,32,48,55,34,47,33,45)(27,36,50,52,37,39,35,49)(31,43,54,56,44,46,42,53)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=g,e^2=fghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6750498);