Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1739 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)(21,22,28,42,27,32,26,33)(23,30,38,29,24,36,35,45)(25,31,43,52,41,48,40,49)(34,46,50,44,37,47,39,51);(7,13)(8,16)(10,17)(11,15)(21,23)(22,29)(24,27)(25,39)(26,38)(28,35)(30,42)(31,46)(32,45)(33,36)(34,40)(37,43)(41,50)(44,49)(47,48)(51,52);(1,2,3,4)(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,24,27,23)(22,30,32,36)(25,34,41,37)(26,38,28,35)(29,33,45,42)(31,47,48,46)(39,43,50,40)(44,52,51,49);(7,13)(10,17)(14,19)(18,20)(21,25,28,43,27,41,26,40)(22,31,42,52,32,48,33,49)(23,34,38,50,24,37,35,39)(29,44,36,47,45,51,30,46);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,32)(23,24)(31,48)(33,42)(34,37)(35,38)(39,50)(49,52)Polycyclic presentation:
2^9|a^2=fh,c^2=ghi,d^2=fi,e^2=hi,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=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7103965);