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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1576 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)(3,5,4,7)(10,12)(11,15)(19,25)(20,27)(23,28)(24,29)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,3)(2,7)(4,6)(5,8)(9,10,13,12)(11,16,15,14)(18,22)(23,28)(24,29)(30,32)(36,44)(39,46)(42,48)(45,47);(3,4)(5,7)(10,12)(11,15)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,4,6,3)(2,7,8,5)(9,11,13,15)(10,14,12,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,5,6,7)(2,3,8,4)(9,12,13,10)(11,16,15,14)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=hi,b^2=fghi,d^2=gh,e^2=f,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, 4976561);