Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,6,8,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(33,34,38,39)(35,42,40,36)(37,41,45,47)(43,48,46,44);(1,3,5,8)(2,6,4,7)(9,10)(11,12)(13,14)(15,16)(18,22)(19,25)(24,28)(27,31)(34,39)(35,40)(37,45)(44,48);(2,4)(6,7)(10,14)(11,15)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32)(33,35,38,40)(34,36,39,42)(37,46,45,43)(41,48,47,44);(1,4,5,2)(3,7,8,6)(9,11,13,15)(10,12,14,16)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,36)(34,40)(35,39)(37,44)(38,42)(41,46)(43,47)(45,48);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,14)(13,16)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)(33,37,38,45)(34,41,39,47)(35,43,40,46)(36,44,42,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=fghi,c^2=i,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6176601);