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 | |
1386 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,25,27,21)(20,28,29,24)(26,30,32,31)(34,38)(35,41)(36,44)(46,48);(4,8)(6,7)(18,23)(20,29)(21,25)(26,32)(33,34,37,38)(35,42,41,39)(36,45,44,40)(43,46,47,48);(2,3)(6,7)(17,19,22,27)(18,21,23,25)(20,26,29,32)(24,30,28,31)(33,34)(35,39)(36,45)(37,38)(40,44)(41,42)(43,48)(46,47);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20,22,29)(18,24,23,28)(19,26,27,32)(21,30,25,31)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,13,14,16)(17,21)(18,19)(20,30)(22,25)(23,27)(24,26)(28,32)(29,31)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=fghi,b^2=h,c^2=fh,d^2=g,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, 5071856);