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 | |
1308 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)(3,4)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(18,22)(19,25)(20,27)(30,32)(33,34)(35,40)(36,43)(37,39)(38,41)(42,47)(44,48)(45,46);(2,6)(3,7)(18,22)(19,25)(24,29)(26,31)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(2,6)(4,8)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,39,35)(34,40,37,43)(38,44,46,42)(41,47,45,48);(1,3,5,7)(2,4,6,8)(9,11,12,16)(10,13,14,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,4)(2,7)(3,6)(5,8)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,38,39,46)(34,41,37,45)(35,42,36,44)(40,47,43,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=fgh,c^2=fgh,d^2=fi,e^2=fg,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, 4581979);