Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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:
(3,7)(6,8)(9,10,13,11)(12,15,16,14)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,34)(35,36)(37,41)(38,39)(40,42)(43,44)(45,47)(46,48);(2,5)(6,8)(9,10,13,11)(12,14,16,15)(20,28)(23,30)(26,32)(29,31)(34,39)(35,42)(37,45)(44,46);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,35)(34,40)(36,39)(37,43)(38,42)(41,46)(44,47)(45,48);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,36,38,40)(34,35,39,42)(37,46,45,44)(41,48,47,43);(1,3)(2,6)(4,7)(5,8)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,37,38,45)(34,41,39,47)(35,43,42,48)(36,44,40,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=fhi,c^2=i,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5841541);