Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(8,16)(11,17)(15,20)(18,19)(21,22,26,31,27,32,25,40)(23,28,35,46,36,47,41,45)(24,37,38,49,39,30,42,29)(33,44,51,48,34,52,50,43);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,36)(22,28,32,47)(24,34,39,33)(25,41,26,35)(29,48,49,43)(30,44,37,52)(31,46,40,45)(38,50,42,51);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,27,39)(22,29,32,49)(23,33,36,34)(25,38,26,42)(28,43,47,48)(30,40,37,31)(35,50,41,51)(44,46,52,45);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,24)(22,30)(23,34)(25,42)(26,38)(27,39)(28,44)(29,31)(32,37)(33,36)(35,50)(40,49)(41,51)(43,45)(46,48)(47,52);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,25,27,26)(22,31,32,40)(23,35,36,41)(24,38,39,42)(28,45,47,46)(29,30,49,37)(33,50,34,51)(43,44,48,52)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7090977);