Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1649 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)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,28,34)(23,30,37,43)(24,29,36,44)(25,31,39,46)(26,33,27,32)(35,45,49,52)(38,42,50,51)(40,48,41,47);(2,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,27,36,28,37,26,24)(22,29,33,43,34,44,32,30)(25,35,41,50,39,49,40,38)(31,42,48,52,46,51,47,45);(7,13)(10,17)(14,15)(18,19)(21,24,26,37,28,36,27,23)(22,30,32,44,34,43,33,29)(25,38,40,49,39,50,41,35)(31,45,47,51,46,52,48,42);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(21,25,28,39)(22,31,34,46)(23,35,37,49)(24,38,36,50)(26,40,27,41)(29,42,44,51)(30,45,43,52)(32,47,33,48);(6,12)(10,17)(11,20)(18,19)(25,39)(31,46)(35,49)(38,50)(40,41)(42,51)(45,52)(47,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6391892);