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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(11,15)(13,16)(17,18)(19,22)(20,25)(21,26)(23,28)(24,30)(27,31)(29,32)(33,34,37,38)(35,39,41,46)(36,40,45,44)(42,47,43,48);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(18,19)(23,25)(24,26)(31,32)(34,38)(35,41)(40,44)(42,43);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,35)(34,39)(36,43)(37,41)(38,46)(40,48)(42,45)(44,47);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,15)(14,16)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,34)(35,39)(36,44)(37,38)(40,45)(41,46)(42,48)(43,47);(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,36,37,45)(34,40,38,44)(35,42,41,43)(39,47,46,48)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=g,d^2=ghi,e^2=f,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, 3408668);