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 | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1642 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)(21,22,26,33,27,32,42,31)(23,34,37,46,38,28,43,47)(24,39,41,48,25,30,40,29)(35,45,52,44,36,50,51,49);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,38)(22,28,32,34)(24,35,25,36)(26,37,42,43)(29,44,48,49)(30,45,39,50)(31,46,33,47)(40,51,41,52);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,27,25)(22,29,32,48)(23,35,38,36)(26,40,42,41)(28,44,34,49)(30,31,39,33)(37,51,43,52)(45,46,50,47);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,25,27,24)(22,30,32,39)(23,36,38,35)(26,40,42,41)(28,45,34,50)(29,31,48,33)(37,51,43,52)(44,46,49,47);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,27,42)(22,31,32,33)(23,37,38,43)(24,40,25,41)(28,46,34,47)(29,39,48,30)(35,51,36,52)(44,50,49,45)Polycyclic presentation:
2^9|a^2=fgi,b^2=hi,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dfi,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, 7091066);