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 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,4,7,8)(9,10,12,14)(11,13,15,16)(18,22)(20,28)(23,25)(26,32)(33,34)(35,40)(36,42)(37,41)(38,39)(43,48)(44,45)(46,47);(2,6)(3,7)(9,11,12,15)(10,13,14,16)(18,22)(20,28)(23,25)(26,32)(34,39)(35,42)(41,47)(43,45);(3,7)(4,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,42)(34,40,39,36)(37,45,46,43)(41,44,47,48);(1,3,5,7)(2,4,6,8)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,40)(34,35,39,42)(37,44,46,48)(41,43,47,45);(1,4)(2,7)(3,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45)Polycyclic presentation:
2^9|a^2=fgh,b^2=ghi,c^2=i,d^2=fg,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, 6176863);