Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574298 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1398 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)(12,16)(13,14)(17,18,22,26)(19,27,29,42)(20,31,33,46)(21,35,32,25)(23,36,38,30)(24,39,40,34)(28,43,44,37)(41,47,48,45);(2,6)(7,8)(10,14)(11,12)(17,19,21,30,22,29,32,36)(18,23,25,27,26,38,35,42)(20,28,34,45,33,44,39,47)(24,37,31,41,40,43,46,48);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,20,32,39,22,33,21,34)(18,24,35,46,26,40,25,31)(19,28,36,47,29,44,30,45)(23,37,42,48,38,43,27,41);(1,4,5,3)(2,8,6,7)(11,12)(15,16)(19,29)(20,33)(23,38)(24,40)(27,42)(30,36)(31,46)(34,39);(9,12,13,11)(10,16,14,15)(20,33)(24,40)(28,44)(31,46)(34,39)(37,43)(41,48)(45,47)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fi,d^2=fghi,e^2=h,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=dhi,e^c=ei,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6532046);