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) = C25×C4 |
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 | |
1386 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)(3,6)(4,5)(7,8)(9,10)(11,12)(13,15)(14,16)(18,22)(23,25)(24,29)(30,31)(33,34,38,42)(35,43,44,39)(36,37,41,40)(45,46,47,48);(2,5)(3,7)(9,11,13,16)(10,14,15,12)(20,28)(24,29)(26,32)(30,31)(35,44)(36,41)(37,40)(39,43);(2,5)(6,8)(10,15)(11,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,35)(34,39)(36,47)(37,46)(38,44)(40,48)(41,45)(42,43);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,11,15,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,41)(34,40,42,37)(35,45,44,47)(39,46,43,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37,38,40)(34,41,42,36)(35,46,44,48)(39,45,43,47)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=fgi,d^2=g,e^2=fg,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, 6189451);