Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470137 |
| 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×#3515 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2862 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,33)(24,38)(25,30)(26,31)(27,32)(28,37)(29,39)(34,48)(35,43)(36,44)(40,46)(41,45)(42,50)(47,51)(49,52);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,27,36)(22,28,32,43)(24,34,41,50)(25,35,26,37)(29,42,46,48)(30,33,31,44)(38,52,45,47)(39,49,40,51);(7,13)(10,17)(14,19)(18,20)(21,24,27,41)(22,29,32,46)(23,34,36,50)(25,39,26,40)(28,42,43,48)(30,45,31,38)(33,47,44,52)(35,49,37,51);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,32,31)(23,35,36,37)(24,39,41,40)(28,33,43,44)(29,45,46,38)(34,49,50,51)(42,47,48,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(23,36)(28,43)(33,44)(34,50)(35,37)(42,48)(47,52)(49,51)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,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, 6403406);