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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2694 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)(21,22,26,32,27,30,25,31)(23,33,37,28,35,45,36,44)(24,38,42,47,40,48,41,29)(34,46,51,52,39,43,50,49);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,27,35)(22,28,30,44)(24,39,40,34)(25,37,26,36)(29,46,47,43)(31,45,32,33)(38,49,48,52)(41,50,42,51);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,27,40)(22,29,30,47)(23,34,35,39)(25,42,26,41)(28,43,44,46)(31,38,32,48)(33,49,45,52)(36,51,37,50);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,35)(24,40)(28,44)(29,47)(33,45)(36,37)(38,48)(41,42);(6,12)(10,13)(11,20)(18,19)(22,30)(23,35)(29,47)(31,32)(34,39)(36,37)(38,48)(50,51)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=hi,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, 6438219);