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 | #2011 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1745 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)(21,22,28,40,27,25,26,31)(23,32,41,29,35,45,34,39)(24,30,42,50,37,36,38,44)(33,46,51,43,47,52,48,49);(7,13)(8,16)(10,17)(11,15)(21,23,27,35)(22,29,25,39)(24,33,37,47)(26,41,28,34)(30,43,36,49)(31,45,40,32)(38,51,42,48)(44,52,50,46);(1,2,3,4)(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,24)(22,30)(23,33)(25,36)(26,38)(27,37)(28,42)(29,43)(31,44)(32,46)(34,48)(35,47)(39,49)(40,50)(41,51)(45,52);(7,13)(10,17)(14,19)(18,20)(21,25,28,31,27,22,26,40)(23,32,41,29,35,45,34,39)(24,36,42,44,37,30,38,50)(33,46,51,43,47,52,48,49);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(24,37)(30,36)(33,47)(38,42)(43,49)(44,50)(46,52)(48,51)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=gh,d^2=fi,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=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7103960);