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 | |
1316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,4,7)(18,22)(20,27)(23,28)(26,31)(33,34,39,37)(35,41,36,40)(38,45,46,42)(43,47,44,48);(10,13)(14,16)(18,22)(19,25)(24,29)(26,31)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,36,39,35)(34,41,37,40)(38,43,46,44)(42,47,45,48);(1,2)(3,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,37,39,34)(35,41,36,40)(38,42,46,45)(43,47,44,48);(1,4,5,3)(2,7,8,6)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fhi,d^2=fh,e^2=gi,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, 3614200);