Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500450 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #37418 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(17,18)(19,26)(20,25)(21,27)(22,28)(23,29)(24,30)(31,42)(32,41)(33,40)(34,39)(35,48)(36,44)(37,45)(38,46)(43,47);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,32,24,33,23,20)(18,25,29,40,30,41,28,26)(21,35,37,34,38,31,36,47)(27,43,44,42,46,39,45,48);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,23,33,24,32,22,19)(18,26,28,41,30,40,29,25)(21,34,36,35,38,47,37,31)(27,42,45,43,46,48,44,39);(11,16)(13,15)(17,21)(18,27)(19,31)(20,34)(22,37)(23,36)(24,38)(25,39)(26,42)(28,45)(29,44)(30,46)(32,47)(33,35)(40,48)(41,43);(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,32,33,20)(21,36,38,37)(25,26,41,40)(27,44,46,45)(31,34,35,47)(39,48,43,42)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6413765);