Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(20,27)(24,29)(25,31)(30,32)(33,34,38,36)(35,41,42,39)(37,40,46,44)(43,48,45,47);(1,3,6,7)(2,4,8,5)(10,13)(14,15)(18,22)(23,28)(24,29)(30,32)(34,36)(35,42)(40,44)(43,45);(1,4,6,5)(2,7,8,3)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35,38,42)(34,39,36,41)(37,45,46,43)(40,48,44,47);(1,5,6,4)(2,7,8,3)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36,38,34)(35,41,42,39)(37,44,46,40)(43,48,45,47);(1,2)(3,4)(5,7)(6,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37,38,46)(34,40,36,44)(35,43,42,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,d^2=fghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4626052);