Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(4,7)(9,10,13,11)(12,15,16,14)(17,18,21,22)(19,23,25,30)(20,27,29,24)(26,32,28,31)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,45)(46,48);(2,6)(3,8)(9,10,13,11)(12,14,16,15)(18,22)(19,25)(24,27)(26,28)(34,36)(39,41)(40,44)(46,47);(1,2)(3,7)(4,8)(5,6)(9,11,13,10)(12,15,16,14)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3,5,8)(2,7,6,4)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,36,38,34)(35,41,43,39)(37,40,45,44)(42,46,48,47);(1,4,5,7)(2,8,6,3)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=fi,b^2=ghi,c^2=i,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4796413);