Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(11,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,34,38,39)(35,36,41,40)(37,45,48,42)(43,47,46,44);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(24,27)(26,28)(34,39)(35,41)(42,45)(43,46);(1,2,4,5)(3,6,7,8)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35,38,41)(34,40,39,36)(37,46,48,43)(42,44,45,47);(9,11,12,15)(10,13,14,16)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31)(33,36,38,40)(34,41,39,35)(37,47,48,44)(42,43,45,46);(1,3,4,7)(2,6,5,8)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,37)(34,42)(35,43)(36,44)(38,48)(39,45)(40,47)(41,46)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fgh,d^2=fi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3414565);