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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,31,32,30)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,45)(46,48);(3,7)(6,8)(9,11,13,16)(10,14,15,12)(18,19)(21,29)(23,25)(27,32)(34,36)(37,45)(39,41)(42,48);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,36,38,34)(35,39,43,41)(37,44,45,40)(42,46,48,47);(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=f,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, 4384362);