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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,4,7)(9,10,13,12)(11,14,16,15)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,32)(30,31)(33,34,38,40)(35,41,43,36)(37,45,46,39)(42,44,48,47);(1,3,5,4)(2,6,8,7)(9,11,13,16)(10,14,12,15)(18,19)(21,29)(23,25)(27,30)(37,46)(39,45)(42,48)(44,47);(9,11)(10,14)(12,15)(13,16)(17,19,22,18)(20,23,28,25)(21,24,29,26)(27,32,30,31)(33,35,38,43)(34,36,40,41)(37,42,46,48)(39,44,45,47);(1,2)(3,6)(4,7)(5,8)(9,10,13,12)(11,14,16,15)(17,20)(18,23)(19,25)(21,30)(22,28)(24,32)(26,31)(27,29)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,4,5,3)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,21,22,29)(18,24,19,26)(20,27,28,30)(23,31,25,32)(33,37,38,46)(34,39,40,45)(35,42,43,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=fhi,c^2=g,d^2=fh,e^2=fghi,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, 4206475);