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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1401 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)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(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)(45,51)(52,55)(54,56);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,31,38)(26,32,35,47)(28,36,42,52)(29,43,30,37)(33,45,49,56)(34,48,44,46)(39,50,53,55)(40,54,41,51);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,28,30,41,31,42,29,40)(26,33,34,39,35,49,44,53)(27,36,37,51,38,52,43,54)(32,45,48,50,47,56,46,55);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(25,29,31,30)(26,34,35,44)(27,37,38,43)(28,40,42,41)(32,46,47,48)(33,39,49,53)(36,51,52,54)(45,55,56,50);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,38)(32,47)(36,52)(37,43)(45,56)(46,48)(50,55)(51,54)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=i,e^2=f,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, 6820399);