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 | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,8)(3,4,5,7)(9,10)(11,15)(12,14)(13,16)(18,22)(19,25)(24,29)(27,31)(33,34)(35,41)(36,38)(37,40)(39,43)(42,48)(44,45)(46,47);(1,3)(2,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(20,28)(24,29)(27,31)(30,32)(34,36)(37,45)(39,41)(42,46);(2,8)(3,5)(9,11,12,16)(10,13,14,15)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35)(34,39)(36,41)(37,46)(38,43)(40,48)(42,45)(44,47);(1,4)(2,3)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,48,46,47);(1,5,6,3)(2,4,8,7)(9,11)(10,13)(12,16)(14,15)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37,38,45)(34,40,36,44)(35,42,43,46)(39,47,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=gi,d^2=fh,e^2=i,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, 5111587);