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 | |
1460 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)(3,6)(4,8)(5,7)(9,10,13,15)(11,12,16,14)(19,25)(20,27)(23,28)(24,29)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,46)(47,48);(1,3)(2,6)(4,7)(5,8)(9,10,13,15)(11,12,16,14)(18,22)(23,28)(24,29)(30,32)(34,37)(39,41)(40,44)(46,47);(2,7)(3,8)(9,11,13,16)(10,14,15,12)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,43)(34,39,37,41)(36,42,45,48)(40,46,44,47);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,34)(35,41,43,39)(36,40,45,44)(42,46,48,47)Polycyclic presentation:
2^9|a^2=fhi,b^2=fhi,c^2=fgi,d^2=gh,e^2=f,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, 5699386);