Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1379 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)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18,21,22)(19,23,25,29)(20,24,28,27)(26,30,31,32)(33,34,38,41)(35,42,39,36)(37,45,46,40)(43,44,47,48);(1,3,5,6)(2,4,7,8)(18,22)(19,25)(24,27)(26,31)(35,39)(36,42)(43,47)(44,48);(1,4,5,8)(2,6,7,3)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(9,11)(10,13)(12,15)(14,16)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,2)(3,4)(5,7)(6,8)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)(33,37,38,46)(34,40,41,45)(35,43,39,47)(36,44,42,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=ghi,d^2=fgh,e^2=g,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, 4546348);