Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,16)(11,12,13,14)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(33,34,38,39)(35,36,40,43)(37,41,46,47)(42,44,45,48);(3,7)(5,8)(9,11,15,13)(10,12,16,14)(19,25)(22,29)(26,31)(30,32)(34,39)(36,43)(41,47)(44,48);(1,2,4,6)(3,5,7,8)(9,12,15,14)(10,11,16,13)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,35)(34,36)(37,45)(38,40)(39,43)(41,48)(42,46)(44,47);(1,2)(3,5)(4,6)(7,8)(9,13,15,11)(10,12,16,14)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36)(34,40)(35,39)(37,44)(38,43)(41,45)(42,47)(46,48);(1,3,4,7)(2,5,6,8)(9,14,15,12)(10,11,16,13)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,37)(34,41)(35,42)(36,44)(38,46)(39,47)(40,45)(43,48)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=gh,d^2=hi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5030380);