Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667980 |
| 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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1265 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)(15,21)(20,24)(22,23)(25,26,31,34,30,35,29,43)(27,36,38,47,39,32,44,48)(28,40,41,50,42,49,45,33)(37,52,55,46,54,56,53,51);(1,2,4,5)(3,7,8,6)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24)(25,27)(26,32)(28,37)(29,44)(30,39)(31,38)(33,46)(34,48)(35,36)(40,56)(41,55)(42,54)(43,47)(45,53)(49,52)(50,51);(2,5)(6,7)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28,30,42)(26,33,35,50)(27,37,39,54)(29,41,31,45)(32,46,36,51)(34,49,43,40)(38,53,44,55)(47,56,48,52);(10,17)(15,19)(16,18)(23,24)(25,29,30,31)(26,34,35,43)(27,38,39,44)(28,41,42,45)(32,47,36,48)(33,49,50,40)(37,53,54,55)(46,56,51,52);(1,3,4,8)(2,6,5,7)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(27,39)(32,36)(37,54)(38,44)(46,51)(47,48)(52,56)(53,55)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6710102);