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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(5,8)(11,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,34,38,39)(35,36,41,44)(37,40,46,47)(42,43,45,48);(1,2,4,6)(3,5,7,8)(10,13)(14,16)(18,19)(20,27)(24,26)(28,32)(34,39)(35,41)(40,47)(42,45);(1,2)(3,5)(4,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,35)(34,36)(37,45)(38,41)(39,44)(40,48)(42,46)(43,47);(9,11)(10,14)(12,15)(13,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,47)(46,48);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,46)(34,40,39,47)(35,42,41,45)(36,43,44,48)Polycyclic presentation:
2^9|a^2=fg,b^2=ghi,c^2=fi,d^2=g,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, 4556588);