Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | #36015 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1739 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,25,29)(23,31,34,27)(24,28,36,44)(26,30,38,46)(32,49,50,41)(33,47,39,42)(35,52,51,43)(37,45,40,48);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,25,34)(22,27,29,31)(24,35,36,51)(26,39,38,33)(28,43,44,52)(30,47,46,42)(32,40,50,37)(41,48,49,45);(1,2,3,4)(21,24,25,36)(22,28,29,44)(23,32,34,50)(26,40,38,37)(27,41,31,49)(30,48,46,45)(33,51,39,35)(42,52,47,43);(6,11)(10,13)(12,19)(18,20)(24,36)(28,44)(32,50)(35,51)(37,40)(41,49)(43,52)(45,48);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,33)(26,38)(27,42)(30,46)(31,47)(32,51)(34,39)(35,50)(37,40)(41,52)(43,49)(45,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593413);