Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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:
(1,2,4,6)(3,7,8,5)(11,15)(13,16)(17,18,22,25)(19,20,26,23)(21,29,30,24)(27,31,32,28)(33,34)(35,40)(36,43)(37,39)(38,41)(42,47)(44,48)(45,46);(1,2)(3,7)(4,6)(5,8)(19,26)(20,23)(27,32)(28,31)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(1,2,4,6)(3,5,8,7)(9,10,12,14)(11,13,15,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,31)(28,29)(30,32)(33,36,39,35)(34,40,37,43)(38,44,46,42)(41,47,45,48);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,13,14,16)(17,20,22,23)(18,19,25,26)(21,28,30,31)(24,27,29,32)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,3,4,8)(2,5,6,7)(17,21,22,30)(18,24,25,29)(19,27,26,32)(20,28,23,31)(33,38,39,46)(34,41,37,45)(35,42,36,44)(40,47,43,48)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=fhi,d^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4430522);