Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,27,32)(23,28,36,44)(24,29,39,47)(25,30,26,31)(33,41,50,52)(34,42,35,43)(37,45,49,51)(38,46,40,48);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23)(22,28)(24,37)(25,34)(26,35)(27,36)(29,45)(30,42)(31,43)(32,44)(33,38)(39,49)(40,50)(41,46)(47,51)(48,52);(2,4)(21,24,27,39)(22,29,32,47)(23,33,36,50)(25,38,26,40)(28,41,44,52)(30,46,31,48)(34,49,35,37)(42,51,43,45);(6,11)(10,13)(12,19)(18,20)(21,25,27,26)(22,30,32,31)(23,34,36,35)(24,38,39,40)(28,42,44,43)(29,46,47,48)(33,49,50,37)(41,51,52,45);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,35)(25,26)(28,43)(30,31)(33,37)(34,36)(38,40)(41,45)(42,44)(46,48)(49,50)(51,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580822);