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 | |
4036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,9,4,8)(5,13,14,10)(11,15,12,16)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30)(33,34,39,42)(35,40,37,45)(36,38,41,48)(43,44,46,47);(2,7)(8,9)(10,13)(15,16)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(35,37)(40,45)(43,46)(44,47);(1,3,6,4)(2,8,7,9)(5,11,14,12)(10,15,13,16)(17,19,22,20)(18,24,26,23)(21,27,29,28)(25,31,32,30)(33,35,39,37)(34,40,42,45)(36,43,41,46)(38,44,48,47);(1,4,6,3)(2,9,7,8)(5,11,14,12)(10,15,13,16)(17,20,22,19)(18,24,26,23)(21,27,29,28)(25,30,32,31)(33,36)(34,38)(35,43)(37,46)(39,41)(40,44)(42,48)(45,47);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18)(19,24)(20,23)(21,25)(22,26)(27,31)(28,30)(29,32)(33,37,39,35)(34,40,42,45)(36,43,41,46)(38,47,48,44);(1,5,6,14)(2,10,7,13)(3,11,4,12)(8,15,9,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,38)(34,41)(35,44)(36,42)(37,47)(39,48)(40,46)(43,45)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gi,d^2=g,e^2=i,f^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, 10347210);