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 | |
4182 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)(3,7,12,5)(4,8,14,16)(10,15,11,13)(17,18,22,26)(19,21,29,25)(20,24,31,23)(27,32,28,30)(33,34)(35,42)(36,45)(37,40)(38,44)(39,41)(43,47)(46,48);(1,3,6,12)(2,7,9,5)(4,11,14,10)(8,13,16,15)(17,19)(18,21)(20,28)(22,29)(23,32)(24,30)(25,26)(27,31)(33,35)(34,38)(36,46)(37,43)(39,44)(40,48)(41,42)(45,47);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,20)(18,23)(19,27)(21,30)(22,31)(24,26)(25,32)(28,29)(33,36,39,37)(34,40,41,45)(35,43,44,46)(38,47,42,48);(1,4)(2,8)(3,11)(5,15)(6,14)(7,13)(9,16)(10,12)(17,20)(18,24)(19,28)(21,30)(22,31)(23,26)(25,32)(27,29)(33,35)(34,38)(36,43)(37,46)(39,44)(40,47)(41,42)(45,48);(1,3)(2,5)(4,10)(6,12)(7,9)(8,13)(11,14)(15,16)(17,21,22,25)(18,19,26,29)(20,30,31,32)(23,27,24,28)(33,37,39,36)(34,40,41,45)(35,43,44,46)(38,48,42,47);(1,5)(2,3)(4,13)(6,7)(8,10)(9,12)(11,16)(14,15)(17,19)(18,25)(20,27)(21,26)(22,29)(23,32)(24,30)(28,31)(33,38)(34,35)(36,47)(37,48)(39,42)(40,43)(41,44)(45,46)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10327514);