Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #53260 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,8)(4,11)(5,13)(6,7)(9,15)(10,12)(14,16)(18,23)(19,25)(26,31)(27,32)(33,34,39,42)(35,37,41,47)(36,38,46,40)(43,48,44,45);(1,3,6,11)(2,4,7,8)(5,10,15,14)(9,12,13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31)(33,35,39,41)(34,37,42,47)(36,44,46,43)(38,45,40,48);(2,7)(4,8)(9,13)(12,16)(33,36,39,46)(34,40,42,38)(35,43,41,44)(37,45,47,48);(1,4,6,8)(2,3,7,11)(5,12,15,16)(9,10,13,14)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,29,31,28)(33,36)(34,38)(35,44)(37,45)(39,46)(40,42)(41,43)(47,48);(1,3)(2,8)(4,7)(5,14)(6,11)(9,12)(10,15)(13,16)(17,20,22,24)(18,25,23,19)(21,29,30,28)(26,27,31,32)(33,37,39,47)(34,41,42,35)(36,45,46,48)(38,43,40,44);(1,5)(2,9)(3,10)(4,12)(6,15)(7,13)(8,16)(11,14)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,38)(34,36)(35,45)(37,44)(39,40)(41,48)(42,46)(43,47)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,d^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9592705);