Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22,26,30)(23,27,34,43)(24,28,36,45)(25,29,39,48)(31,40,49,51)(32,41,33,42)(35,44,50,52)(37,46,38,47);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23)(22,27)(24,35)(25,32)(26,34)(28,44)(29,41)(30,43)(31,37)(33,39)(36,50)(38,49)(40,46)(42,48)(45,52)(47,51);(2,4)(21,24)(22,28)(23,31)(25,38)(26,36)(27,40)(29,47)(30,45)(32,35)(33,50)(34,49)(37,39)(41,44)(42,52)(43,51)(46,48);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(24,36)(28,45)(31,49)(35,50)(37,38)(40,51)(44,52)(46,47);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,32)(25,39)(27,41)(29,48)(31,35)(33,34)(37,38)(40,44)(42,43)(46,47)(49,50)(51,52)Polycyclic presentation:
2^9|a^2=fi,b^2=f,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6584474);