Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(9,10,13,11)(12,15,16,14)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30)(33,34,38,39)(35,36,42,40)(37,45,46,41)(43,48,47,44);(2,6)(7,8)(9,11,13,10)(12,15,16,14)(20,29)(23,28)(26,32)(30,31)(34,39)(35,42)(37,46)(44,48);(1,2)(3,7)(4,8)(5,6)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,35,38,42)(34,40,39,36)(37,47,46,43)(41,44,45,48);(1,3,5,4)(2,7,6,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,36,38,40)(34,35,39,42)(37,48,46,44)(41,47,45,43);(1,4,5,3)(2,8,6,7)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,48)(42,47)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=f,d^2=h,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4115360);