Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907909 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,32)(28,39)(29,34)(30,44)(31,35)(33,41)(36,50)(37,46)(38,47)(40,49)(42,53)(43,48)(45,52)(51,56)(54,55);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,37)(26,32,35,46)(28,36,42,51)(29,43,30,38)(33,45,49,55)(34,48,44,47)(39,50,53,56)(40,54,41,52);(1,3)(2,5)(4,8)(6,7)(12,21)(15,20)(18,24)(22,23)(25,28,29,40,31,42,30,41)(26,33,44,53,35,49,34,39)(27,36,43,54,37,51,38,52)(32,45,47,56,46,55,48,50);(3,8)(5,7)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(27,37)(32,46)(36,51)(38,43)(45,55)(47,48)(50,56)(52,54);(2,6)(5,7)(11,19)(14,17)(18,24)(22,23)(25,29,31,30)(26,34,35,44)(27,38,37,43)(28,40,42,41)(32,47,46,48)(33,39,49,53)(36,52,51,54)(45,56,55,50)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6817052);