Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | C2×#496 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1764 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,27,28,26,32,25,31)(23,29,38,40,34,43,35,44)(24,30,39,41,37,45,36,46)(33,42,49,50,48,52,47,51);(7,13)(8,16)(10,17)(11,15)(22,28)(24,36)(25,27)(29,40)(31,32)(33,47)(35,38)(37,39)(41,46)(43,44)(48,49)(50,51);(1,2,3,4)(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,23,26,34)(22,29,32,43)(24,33,37,48)(25,35,27,38)(28,40,31,44)(30,42,45,52)(36,47,39,49)(41,50,46,51);(7,13)(10,17)(14,19)(18,20)(21,24,25,36,26,37,27,39)(22,30,31,46,32,45,28,41)(23,33,35,47,34,48,38,49)(29,42,44,51,43,52,40,50);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(23,34)(24,37)(29,43)(30,45)(35,38)(36,39)(40,44)(41,46)Polycyclic presentation:
2^9|a^2=fhi,c^2=ghi,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7103990);