Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22,27,32)(23,28,34,42)(24,37,39,46)(25,31,26,30)(29,40,45,38)(33,47,48,52)(35,44,36,43)(41,50,51,49);(1,2)(3,4)(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23)(22,28)(24,33)(25,35)(26,36)(27,34)(29,41)(30,43)(31,44)(32,42)(37,47)(38,49)(39,48)(40,50)(45,51)(46,52);(2,4)(21,24,27,39)(22,29,32,45)(23,33,34,48)(25,38,26,40)(28,41,42,51)(30,37,31,46)(35,49,36,50)(43,47,44,52);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(23,34)(28,42)(33,48)(35,36)(41,51)(43,44)(47,52)(49,50);(2,4)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,25,27,26)(22,30,32,31)(23,35,34,36)(24,38,39,40)(28,43,42,44)(29,37,45,46)(33,49,48,50)(41,47,51,52)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6584933);