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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1636 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)(21,22,28,42,27,32,26,31)(23,33,25,29,36,44,35,45)(24,37,39,47,41,30,40,46)(34,49,38,52,51,48,50,43);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,36)(22,29,32,45)(24,34,41,51)(25,28,35,26)(30,43,37,52)(31,44,42,33)(38,39,50,40)(46,48,47,49);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,30)(23,34)(25,38)(26,40)(27,41)(28,39)(29,43)(31,47)(32,37)(33,48)(35,50)(36,51)(42,46)(44,49)(45,52);(7,14)(10,13)(15,20)(18,19)(21,25,26,23,27,35,28,36)(22,29,31,33,32,45,42,44)(24,38,40,34,41,50,39,51)(30,43,47,48,37,52,46,49);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,27,28)(22,31,32,42)(23,25,36,35)(24,39,41,40)(29,44,45,33)(30,46,37,47)(34,50,51,38)(43,48,52,49)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,d^2=f,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=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7091142);