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 | |
1452 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)(3,7)(10,14)(11,16)(18,22)(19,25)(24,29)(27,31)(33,34,38,39)(35,36,41,40)(37,42,45,47)(43,44,48,46);(1,2)(3,8)(4,5)(6,7)(9,10,13,14)(11,15,16,12)(20,28)(24,29)(27,31)(30,32)(34,39)(35,41)(37,45)(44,46);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,15,14,12)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35,38,41)(34,40,39,36)(37,43,45,48)(42,46,47,44);(1,2)(3,8)(4,5)(6,7)(9,12)(10,11)(13,15)(14,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,38,40)(34,41,39,35)(37,46,45,44)(42,43,47,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,42)(35,43)(36,44)(38,45)(39,47)(40,46)(41,48)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=fi,d^2=fgh,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, 5666543);