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×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(8,16)(11,17)(15,20)(18,19)(21,22)(23,28)(24,34)(25,30)(26,40)(27,31)(29,36)(32,44)(33,41)(35,48)(37,43)(38,45)(39,46)(42,49)(47,51)(50,52);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,26)(22,28,31,40)(24,32,37,36)(25,33,39,38)(29,34,44,43)(30,41,46,45)(35,47,50,49)(42,48,51,52);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,23,32,27,37,26,36)(22,29,40,43,31,44,28,34)(25,35,33,47,39,50,38,49)(30,42,45,52,46,51,41,48);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,25,27,39)(22,30,31,46)(23,33,26,38)(24,35,37,50)(28,41,40,45)(29,42,44,51)(32,47,36,49)(34,48,43,52);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,23,27,26)(22,28,31,40)(24,36,37,32)(25,38,39,33)(29,43,44,34)(30,45,46,41)(35,47,50,49)(42,48,51,52)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=gi,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,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, 7093980);