Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500434 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1749 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)(21,22,27,31)(23,32,36,47)(24,29,37,43)(25,30,26,40)(28,34,42,35)(33,46,48,52)(38,44,39,45)(41,50,51,49);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,26,35,27,36,25,34)(22,28,30,32,31,42,40,47)(24,33,38,49,37,48,39,50)(29,41,45,46,43,51,44,52);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24)(22,29)(23,33)(25,39)(26,38)(27,37)(28,41)(30,45)(31,43)(32,46)(34,50)(35,49)(36,48)(40,44)(42,51)(47,52);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(24,37)(29,43)(33,48)(38,39)(41,51)(44,45)(46,52)(49,50);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,38,37,39)(28,32,42,47)(29,44,43,45)(33,49,48,50)(41,52,51,46)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6437123);