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 | |
4095 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,11,5)(4,8,12,15)(10,13,16,14)(17,18,22,25)(19,20,23,26)(21,24,30,32)(27,28,31,29)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(1,3,6,11)(2,5,9,7)(4,10,12,16)(8,14,15,13)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,35)(34,39)(36,38)(37,43)(40,47)(41,42)(44,46)(45,48);(4,12)(8,15)(10,16)(13,14)(19,23)(20,26)(27,31)(28,29)(33,34,38,41)(35,39,36,42)(37,40,46,45)(43,47,44,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,16)(14,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,29)(28,31)(30,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,4,6,12)(2,8,9,15)(3,10,11,16)(5,14,7,13)(17,20,22,26)(18,23,25,19)(21,29,30,28)(24,27,32,31)(33,36,38,35)(34,39,41,42)(37,43,46,44)(40,48,45,47);(1,5,6,7)(2,3,9,11)(4,13,12,14)(8,16,15,10)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,32)(26,29)(33,37,38,46)(34,40,41,45)(35,43,36,44)(39,47,42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9962973);