Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10,13,12)(11,14,16,15)(18,22)(23,25)(24,29)(30,31)(36,43)(39,46)(41,47)(44,48);(9,10)(11,15)(12,13)(14,16)(20,28)(24,29)(26,32)(30,31)(33,34,37,40)(35,38,42,45)(36,39,43,46)(41,44,47,48);(9,11,13,16)(10,14,12,15)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,2)(3,5)(4,6)(7,8)(9,12,13,10)(11,14,16,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,35,37,42)(34,38,40,45)(36,41,43,47)(39,44,46,48);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,14)(13,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48)Polycyclic presentation:
2^9|a^2=fi,b^2=fh,c^2=fg,d^2=gh,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5075026);