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 | |
4027 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)(19,24)(20,22)(21,28)(23,27)(25,29)(26,30)(31,32)(33,34,39,36)(35,41,37,40)(38,44,46,42)(43,47,45,48);(1,3,6,12)(2,7,9,5)(4,11,14,10)(8,13,16,15)(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30)(33,34,39,36)(35,41,37,40)(38,42,46,44)(43,48,45,47);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,4)(2,8)(3,11)(5,15)(6,14)(7,13)(9,16)(10,12)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,36,39,34)(35,41,37,40)(38,42,46,44)(43,47,45,48);(1,3)(2,5)(4,10)(6,12)(7,9)(8,13)(11,14)(15,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,5)(2,3)(4,13)(6,7)(8,10)(9,12)(11,16)(14,15)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gh,d^2=h,e^2=gh,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, 10329882);