Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,4,8,7)(11,15)(14,16)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34)(35,39)(36,40)(37,45)(38,42)(41,46)(43,44)(47,48);(2,6)(4,7)(10,13)(11,15)(18,19)(21,29)(23,25)(27,32)(33,35,38,40)(34,39,42,36)(37,43,46,48)(41,47,45,44);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,47)(46,48);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36)(34,40)(35,42)(37,44)(38,39)(41,48)(43,45)(46,47);(1,4,5,7)(2,3,6,8)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37,38,46)(34,41,42,45)(35,43,40,48)(36,44,39,47)Polycyclic presentation:
2^9|a^2=fghi,b^2=fg,c^2=ghi,d^2=ghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6029430);