Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19)(21,22,26,28)(23,32,33,46)(24,37,38,30)(25,31,27,41)(29,35,42,36)(34,49,48,47)(39,45,40,44)(43,50,52,51);(1,2)(3,4)(5,7,10,17)(6,11,13,15)(8,16,9,18)(12,19,14,20)(22,28)(23,33)(24,38)(31,41)(35,36)(39,40)(43,52)(47,49);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(21,23,26,33)(22,29,28,42)(24,34,38,48)(25,36,27,35)(30,43,37,52)(31,46,41,32)(39,51,40,50)(44,47,45,49);(6,13)(11,15)(12,14)(19,20)(21,24,26,38)(22,30,28,37)(23,34,33,48)(25,40,27,39)(29,43,42,52)(31,45,41,44)(32,47,46,49)(35,51,36,50);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(21,25,26,27)(22,31,28,41)(23,35,33,36)(24,39,38,40)(29,32,42,46)(30,44,37,45)(34,50,48,51)(43,49,52,47)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6582426);