Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,24,30)(19,25,33,42)(20,26,37,46)(21,27,23,29)(22,28,31,40)(32,41,39,48)(34,43,38,47)(35,44,36,45);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,24,33)(18,25,30,42)(20,34,37,38)(21,32,23,39)(22,36,31,35)(26,43,46,47)(27,41,29,48)(28,45,40,44);(10,14)(13,15)(17,20,24,37)(18,26,30,46)(19,31,33,22)(21,36,23,35)(25,40,42,28)(27,45,29,44)(32,38,39,34)(41,47,48,43);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,24,23)(18,27,30,29)(19,32,33,39)(20,35,37,36)(22,38,31,34)(25,41,42,48)(26,44,46,45)(28,47,40,43);(2,5)(6,8)(11,16)(13,15)(17,22,24,31)(18,28,30,40)(19,20,33,37)(21,38,23,34)(25,26,42,46)(27,47,29,43)(32,35,39,36)(41,44,48,45)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6606399);