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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18)(19,26)(20,24)(21,29)(22,23)(25,31)(27,28)(30,32)(34,37)(39,41)(40,45)(46,47);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,31)(30,32)(36,44)(40,45)(42,48)(46,47);(2,9)(3,4)(10,16)(11,12)(18,23)(19,24)(25,29)(27,32)(33,34,38,37)(35,41,43,39)(36,40,44,45)(42,47,48,46);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,19,22,24)(18,20,23,26)(21,27,31,32)(25,28,29,30)(33,35,38,43)(34,39,37,41)(36,42,44,48)(40,46,45,47);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,20)(18,24)(19,23)(21,30)(22,26)(25,27)(28,31)(29,32)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,21)(18,25)(19,27)(20,28)(22,31)(23,29)(24,32)(26,30)(33,37,38,34)(35,41,43,39)(36,45,44,40)(42,47,48,46)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=hi,d^2=h,f^2=hi,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, 9620898);