Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| 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 | |
1308 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)(6,8)(9,10,12,14)(11,13,15,16)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34)(35,39)(36,38)(37,40)(41,46)(42,43)(44,45)(47,48);(9,11,12,15)(10,13,14,16)(18,22)(23,25)(24,28)(30,31)(34,36)(39,43)(40,44)(46,48);(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,42)(34,39,36,43)(37,41,45,47)(40,46,44,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,36,38,34)(35,39,42,43)(37,44,45,40)(41,46,47,48);(1,3,4,7)(2,6,5,8)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,37,38,45)(34,40,36,44)(35,41,42,47)(39,46,43,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fh,d^2=f,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4372352);