Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3307 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2045 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)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,27,31)(23,32,36,46)(24,29,37,43)(25,30,26,40)(28,34,42,35)(33,45,47,52)(38,44,39,50)(41,48,51,49);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,35,27,36,25,34)(22,28,30,32,31,42,40,46)(24,33,39,49,37,47,38,48)(29,41,44,45,43,51,50,52);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,33)(25,38)(26,39)(27,37)(28,41)(30,44)(31,43)(32,45)(34,48)(35,49)(36,47)(40,50)(42,51)(46,52);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(24,37)(29,43)(33,47)(38,39)(41,51)(44,50)(45,52)(48,49);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,38,37,39)(28,32,42,46)(29,44,43,50)(33,48,47,49)(41,45,51,52)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6476938);