Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,8,7,4)(9,10,13,15)(11,16,12,14)(18,22)(19,25)(20,27)(30,32)(35,41)(36,43)(38,44)(39,46);(2,6)(3,7)(9,10)(11,14)(12,16)(13,15)(18,22)(19,25)(24,29)(26,31)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(2,6)(4,8)(9,11,13,12)(10,14,15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,3,5,7)(2,4,6,8)(9,12,13,11)(10,14,15,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,35,37,41)(34,38,40,44)(36,42,43,47)(39,45,46,48);(1,4)(2,7)(3,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,36)(34,39)(35,42)(37,43)(38,45)(40,46)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,c^2=ghi,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4546334);