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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3741 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)(3,6)(4,7)(5,8)(9,10)(11,16)(12,15)(13,14)(17,18,24,27)(19,28,21,22)(20,29,30,25)(23,31,32,26)(33,34,38,41)(35,42,45,39)(36,37,40,46)(43,48,47,44);(1,3)(2,6)(4,10)(5,13)(7,9)(8,14)(11,15)(12,16)(17,19,24,21)(18,22,27,28)(20,26,30,31)(23,29,32,25)(33,35,38,45)(34,39,41,42)(36,43,40,47)(37,44,46,48);(2,4)(3,9)(8,12)(11,13)(17,20,24,30)(18,25,27,29)(19,26,21,31)(22,23,28,32);(1,4,7,2)(3,10,9,6)(5,12,15,8)(11,14,13,16)(17,21,24,19)(18,22,27,28)(20,26,30,31)(23,25,32,29);(1,5,7,15)(2,8,4,12)(3,11,9,13)(6,16,10,14)(17,22)(18,19)(20,23)(21,27)(24,28)(25,26)(29,31)(30,32)(33,36)(34,37)(35,43)(38,40)(39,44)(41,46)(42,48)(45,47);(1,6,7,10)(2,9,4,3)(5,14,15,16)(8,11,12,13)(17,23)(18,26)(19,29)(20,28)(21,25)(22,30)(24,32)(27,31)(33,37)(34,40)(35,44)(36,41)(38,46)(39,47)(42,43)(45,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=hi,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,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, 9620902);