Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
1370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,3)(4,7,8,6)(11,14)(12,16)(20,27)(24,29)(25,31)(30,32)(33,34)(35,41)(36,42)(37,45)(38,39)(40,47)(43,44)(46,48);(9,10,13,15)(11,12,14,16)(18,22)(23,28)(24,29)(30,32)(34,39)(35,42)(40,45)(43,48);(1,3,5,2)(4,7,8,6)(9,11,13,14)(10,12,15,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35)(34,36)(37,43)(38,42)(39,41)(40,44)(45,46)(47,48);(1,4)(2,6)(3,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36)(34,35)(37,46)(38,41)(39,42)(40,48)(43,45)(44,47);(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37,38,47)(34,40,39,45)(35,43,42,48)(36,44,41,46)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=fi,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4340021);