Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884969 |
| 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 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ 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:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,21)(13,17)(14,18)(15,20)(19,23)(22,24)(25,26,31,36)(27,37,40,50)(28,33,41,47)(29,35,30,34)(32,44,46,39)(38,51,52,56)(42,49,43,48)(45,53,55,54);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,19)(13,22)(14,20)(16,18)(17,23)(21,24)(28,41)(33,47)(38,52)(42,43)(45,55)(48,49)(51,56)(53,54);(3,8)(5,7)(9,11)(10,16)(12,22)(13,19)(14,20)(15,18)(17,23)(21,24)(25,27,31,40)(26,32,36,46)(28,38,41,52)(29,39,30,44)(33,45,47,55)(34,37,35,50)(42,53,43,54)(48,56,49,51);(2,6)(5,7)(9,12,14,13)(10,17,18,21)(11,19,20,22)(15,23,16,24)(25,28)(26,33)(27,38)(29,43)(30,42)(31,41)(32,45)(34,49)(35,48)(36,47)(37,51)(39,54)(40,52)(44,53)(46,55)(50,56);(9,13,14,12)(10,17,18,21)(11,19,20,22)(15,24,16,23)(25,29,31,30)(26,34,36,35)(27,39,40,44)(28,42,41,43)(32,37,46,50)(33,48,47,49)(38,53,52,54)(45,56,55,51)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6893484);