Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1713 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,25,29)(23,31,33,27)(24,35,38,28)(26,30,39,44)(32,45,46,40)(34,41,37,43)(36,49,48,42)(47,51,50,52);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(23,32)(24,36)(26,39)(27,40)(28,42)(30,44)(31,45)(33,46)(34,37)(35,49)(38,48)(41,43);(2,4)(21,23,26,32,25,33,39,46)(22,27,30,40,29,31,44,45)(24,37,36,50,38,34,48,47)(28,43,42,52,35,41,49,51);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(23,33)(27,31)(32,46)(34,37)(40,45)(41,43)(47,50)(51,52);(7,15)(10,13)(14,20)(18,19)(21,24,26,36,25,38,39,48)(22,28,30,42,29,35,44,49)(23,34,32,47,33,37,46,50)(27,41,40,51,31,43,45,52)Polycyclic presentation:
2^9|a^2=fi,c^2=h,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715837);