Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,31,35)(27,36,39,47)(28,33,42,52)(29,34,30,45)(32,43,48,38)(37,53,40,56)(41,50,44,51)(46,54,49,55);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22);(2,5)(6,8)(25,27,29,43,31,39,30,38)(26,32,45,47,35,48,34,36)(28,40,44,55,42,37,41,54)(33,49,50,53,52,46,51,56);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(25,28)(26,33)(27,37)(29,44)(30,41)(31,42)(32,46)(34,51)(35,52)(36,53)(38,55)(39,40)(43,54)(45,50)(47,56)(48,49);(2,5)(6,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,29,31,30)(26,34,35,45)(27,38,39,43)(28,41,42,44)(32,47,48,36)(33,50,52,51)(37,54,40,55)(46,53,49,56)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,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, 6694965);