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 | |
1482 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)(3,7,8,4)(9,10)(11,14)(12,16)(13,15)(18,22)(19,25)(20,28)(30,32)(33,34)(35,41)(36,44)(37,38)(39,42)(40,45)(43,47)(46,48);(2,6)(3,8)(9,11,13,16)(10,12,15,14)(19,25)(20,28)(23,26)(24,29)(34,38)(35,42)(36,45)(47,48);(3,8)(4,7)(11,16)(12,14)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35)(34,39)(36,46)(37,42)(38,41)(40,48)(43,45)(44,47);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,45)(46,48);(1,4,5,7)(2,3,6,8)(9,12,13,14)(10,11,15,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,47)(41,48)(42,46)Polycyclic presentation:
2^9|a^2=fh,b^2=ghi,c^2=i,e^2=fg,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, 6176721);