Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,12)(13,14)(15,16)(20,27)(24,29)(25,31)(30,32)(33,34,38,36)(35,41,42,39)(37,44,45,40)(43,46,48,47);(1,3,4,8)(2,5,7,6)(10,14)(11,16)(18,22)(23,28)(24,29)(30,32)(35,42)(37,45)(39,41)(40,44);(1,2)(3,6)(4,7)(5,8)(9,11,13,16)(10,15,14,12)(17,18,21,22)(19,23,26,28)(20,24,27,29)(25,30,31,32)(33,35)(34,39)(36,41)(37,43)(38,42)(40,46)(44,47)(45,48);(1,2)(3,5)(4,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36,38,34)(35,39,42,41)(37,44,45,40)(43,46,48,47);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=fghi,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5700176);