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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(8,16)(11,17)(15,20)(18,19)(21,22,26,30,27,31,25,38)(23,32,34,43,35,42,40,28)(24,29,37,46,36,44,39,45)(33,47,50,51,48,52,49,41);(1,2,3,4)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(24,36)(29,44)(33,48)(37,39)(41,51)(45,46)(47,52)(49,50);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23,27,35)(22,28,31,43)(24,33,36,48)(25,34,26,40)(29,41,44,51)(30,42,38,32)(37,49,39,50)(45,47,46,52);(2,4)(7,14)(10,13)(15,20)(18,19)(21,24)(22,29)(23,33)(25,39)(26,37)(27,36)(28,41)(30,46)(31,44)(32,47)(34,50)(35,48)(38,45)(40,49)(42,52)(43,51);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,25,27,26)(22,30,31,38)(23,34,35,40)(24,37,36,39)(28,42,43,32)(29,45,44,46)(33,49,48,50)(41,47,51,52)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=i,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, 7089786);