Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1633 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)(8,16)(11,17)(15,20)(18,19)(21,22,28,45,27,33,26,32)(23,34,37,29,39,48,38,49)(24,40,41,47,36,30,42,50)(25,43,46,51,44,31,35,52);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,39)(22,29,33,49)(24,35,36,46)(25,41,44,42)(26,37,28,38)(30,31,40,43)(32,48,45,34)(47,52,50,51);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,30)(23,35)(25,37)(26,42)(27,36)(28,41)(29,31)(32,47)(33,40)(34,51)(38,44)(39,46)(43,49)(45,50)(48,52);(7,14)(10,13)(15,20)(18,19)(21,25,28,46,27,44,26,35)(22,31,45,52,33,43,32,51)(23,36,37,42,39,24,38,41)(29,47,48,30,49,50,34,40);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,27,28)(22,32,33,45)(23,37,39,38)(24,41,36,42)(25,35,44,46)(29,48,49,34)(30,50,40,47)(31,51,43,52)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,d^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7090851);