Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500076 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3565 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2762 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)(21,22,25,41,27,31,26,30)(23,32,35,28,34,48,36,44)(24,37,42,29,39,47,40,46)(33,43,51,49,38,45,50,52);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23)(22,28)(24,38)(25,36)(26,35)(27,34)(29,45)(30,48)(31,44)(32,41)(33,39)(37,52)(40,50)(42,51)(43,46)(47,49);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,27,39)(22,29,31,46)(23,33,34,38)(25,40,26,42)(28,43,44,45)(30,47,41,37)(32,49,48,52)(35,50,36,51);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,34)(24,39)(28,44)(29,46)(32,48)(35,36)(37,47)(40,42);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,30,31,41)(23,35,34,36)(24,40,39,42)(28,32,44,48)(29,47,46,37)(33,50,38,51)(43,49,45,52)Polycyclic presentation:
2^9|a^2=fi,c^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436753);