Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,24,30)(19,25,33,41)(20,26,31,39)(21,27,23,29)(22,28,38,46)(32,40,47,48)(34,42,36,44)(35,43,37,45);(9,11)(10,13)(12,16)(14,15)(17,19,24,33)(18,25,30,41)(20,21,31,23)(22,36,38,34)(26,27,39,29)(28,44,46,42)(32,35,47,37)(40,43,48,45);(2,6)(7,8)(10,14)(11,16)(17,20,23,19,24,31,21,33)(18,26,29,25,30,39,27,41)(22,36,37,47,38,34,35,32)(28,44,45,48,46,42,43,40);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,21,24,23)(18,27,30,29)(19,31,33,20)(22,35,38,37)(25,39,41,26)(28,43,46,45)(32,36,47,34)(40,44,48,42);(1,4,5,3)(2,8,6,7)(17,22,23,37,24,38,21,35)(18,28,29,45,30,46,27,43)(19,32,31,36,33,47,20,34)(25,40,39,44,41,48,26,42)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fh,d^2=fgi,e^2=gh,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=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6749195);