Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10,13,12)(11,14,15,16)(17,18)(19,23)(20,22)(21,24)(25,27)(26,31)(28,30)(29,32)(34,38)(35,41)(36,43)(47,48);(11,15)(14,16)(18,20)(23,25)(24,28)(31,32)(33,34,37,38)(35,39,41,46)(36,40,43,44)(42,47,45,48);(1,2,5,4)(3,6,8,7)(9,11,13,15)(10,14,12,16)(17,19,22,27)(18,23,20,25)(21,29,30,26)(24,32,28,31)(33,34)(35,39)(36,44)(37,38)(40,43)(41,46)(42,48)(45,47);(1,3,5,8)(2,6,4,7)(17,20,22,18)(19,25,27,23)(21,24,30,28)(26,31,29,32)(33,35,37,41)(34,39,38,46)(36,45,43,42)(40,48,44,47);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,16,15,14)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,36)(34,40)(35,42)(37,43)(38,44)(39,47)(41,45)(46,48)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,c^2=hi,d^2=i,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3754586);