Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| 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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1353 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)(11,19)(12,16)(13,18)(14,17)(15,22)(20,23)(21,24)(25,26)(27,32)(28,39)(29,34)(30,43)(31,35)(33,40)(36,51)(37,46)(38,47)(41,49)(42,48)(44,50)(45,54)(52,55)(53,56);(1,2)(3,7)(4,6)(5,8)(9,11,14,22)(10,15,17,19)(12,20,18,24)(13,23,16,21)(25,27,31,38)(26,32,35,47)(28,36,41,53)(29,42,30,37)(33,45,50,55)(34,48,43,46)(39,51,49,56)(40,54,44,52);(1,3)(2,5)(4,8)(6,7)(9,12,14,18)(10,16,17,13)(11,20,22,24)(15,21,19,23)(25,28,31,41)(26,33,35,50)(27,36,38,53)(29,40,30,44)(32,45,47,55)(34,49,43,39)(37,52,42,54)(46,51,48,56);(3,8)(5,7)(10,17)(12,18)(15,19)(20,24)(25,29,31,30)(26,34,35,43)(27,37,38,42)(28,40,41,44)(32,46,47,48)(33,49,50,39)(36,52,53,54)(45,51,55,56);(2,6)(5,7)(9,13,14,16)(10,18,17,12)(11,21,22,23)(15,20,19,24)(27,38)(32,47)(36,53)(37,42)(45,55)(46,48)(51,56)(52,54)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,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=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6798291);