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 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10,13,15)(11,12,14,16)(18,22)(19,25)(20,27)(30,32)(33,34,39,41)(35,42,36,37)(38,40,47,48)(43,46,44,45);(9,11)(10,12)(13,14)(15,16)(19,25)(23,28)(26,31)(30,32)(33,35,39,36)(34,37,41,42)(38,43,47,44)(40,45,48,46);(1,2,5,4)(3,6,8,7)(9,12)(10,14)(11,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,39,35)(34,37,41,42)(38,43,47,44)(40,46,48,45);(1,3,5,8)(2,6,4,7)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,37)(34,35)(36,41)(38,46)(39,42)(40,44)(43,48)(45,47);(1,4,5,2)(3,7,8,6)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,38)(34,40)(35,43)(36,44)(37,45)(39,47)(41,48)(42,46)Polycyclic presentation:
2^9|a^2=fghi,b^2=f,c^2=fgi,d^2=i,e^2=i,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, 3733648);