Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ 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)(3,4)(5,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,21,22)(19,23,25,29)(20,27,28,24)(26,32,31,30)(33,34,38,42)(35,39,40,36)(37,41,45,48)(43,46,47,44);(3,8)(4,7)(9,11,13,16)(10,14,15,12)(18,22)(19,25)(20,28)(30,32)(35,40)(36,39)(43,47)(44,46);(2,6)(3,8)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31)(33,35,38,40)(34,39,42,36)(37,43,45,47)(41,46,48,44);(1,3,5,8)(2,7,6,4)(9,11)(10,12)(13,16)(14,15)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32)(33,36,38,39)(34,40,42,35)(37,44,45,46)(41,47,48,43);(1,4,5,7)(2,8,6,3)(9,12)(10,11)(13,14)(15,16)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,37,38,45)(34,41,42,48)(35,43,40,47)(36,44,39,46)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=fh,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6121239);