Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10,13,12)(11,14,16,15)(18,22)(20,28)(23,25)(26,32)(33,34,38,39)(35,36,42,40)(37,45,46,41)(43,48,47,44);(1,2,5,4)(3,6,7,8)(9,11,13,16)(10,14,12,15)(18,22)(20,28)(23,25)(26,32)(34,39)(35,42)(37,46)(44,48);(9,11)(10,14)(12,15)(13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,42)(34,40,39,36)(37,47,46,43)(41,44,45,48);(1,3)(2,6)(4,8)(5,7)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,40)(34,35,39,42)(37,48,46,44)(41,47,45,43);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,15,16,14)(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)(39,45)(40,48)(42,47)Polycyclic presentation:
2^9|a^2=fgh,b^2=fghi,c^2=f,d^2=g,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4191256);