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 | |
1364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(5,8)(9,10,13,15)(11,14,12,16)(17,18,21,22)(19,23,25,26)(20,28,29,24)(27,32,31,30)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,45)(46,48);(9,11,13,12)(10,14,15,16)(18,22)(19,25)(20,29)(30,32)(34,36)(37,45)(39,41)(42,48);(1,2,4,6)(3,5,7,8)(9,12,13,11)(10,14,15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,14)(13,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,36,38,34)(35,39,43,41)(37,44,45,40)(42,46,48,47);(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=i,d^2=fh,e^2=f,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, 4394967);