Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
1721 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,25,29)(23,31,33,27)(24,35,36,44)(26,39,38,30)(28,40,43,37)(32,47,48,52)(34,42,50,46)(41,49,45,51);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,25,33)(22,27,29,31)(24,32,36,48)(26,34,38,50)(28,41,43,45)(30,42,39,46)(35,52,44,47)(37,49,40,51);(2,4)(21,24,25,36)(22,28,29,43)(23,32,33,48)(26,40,38,37)(27,41,31,45)(30,44,39,35)(34,51,50,49)(42,47,46,52);(6,11)(10,13)(12,19)(18,20)(22,29)(23,33)(28,43)(30,39)(32,48)(34,50)(35,44)(49,51);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,33)(24,36)(27,31)(28,43)(34,50)(35,44)(37,40)(42,46)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6581093);