Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(11,15)(14,16)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31)(33,34,37,41)(35,39,43,38)(36,40,45,48)(42,47,44,46);(2,5)(6,8)(10,13)(11,15)(19,26)(21,29)(23,25)(24,28)(33,35,37,43)(34,38,41,39)(36,42,45,44)(40,46,48,47);(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32)(33,35)(34,39)(36,44)(37,43)(38,41)(40,46)(42,45)(47,48);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,47)(45,48);(1,3)(2,6)(4,7)(5,8)(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fgh,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6203899);