Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500076 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2800 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)(7,13)(10,14)(15,16)(18,19)(21,22,27,30,26,32,25,31)(23,33,36,28,35,45,37,43)(24,38,39,29,41,46,40,44)(34,42,50,47,48,51,49,52);(2,4)(5,6)(7,14)(8,11)(9,12)(10,13)(15,19)(16,18)(17,20)(21,23,26,35)(22,28,32,43)(24,34,41,48)(25,36,27,37)(29,42,44,51)(30,33,31,45)(38,47,46,52)(39,49,40,50);(5,7,9,13)(6,10,12,14)(8,16,17,15)(11,19,20,18)(21,24)(22,29)(23,34)(25,39)(26,41)(27,40)(28,42)(30,38)(31,46)(32,44)(33,47)(35,48)(36,49)(37,50)(43,51)(45,52);(5,8,9,17)(6,11,12,20)(7,15,13,16)(10,18,14,19)(23,35)(28,43)(33,45)(34,48)(36,37)(42,51)(47,52)(49,50);(6,12)(10,14)(11,20)(18,19)(21,25,26,27)(22,30,32,31)(23,36,35,37)(24,39,41,40)(28,33,43,45)(29,38,44,46)(34,49,48,50)(42,47,51,52)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=h,d^2=h,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, 6439786);