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 |
| prime radical | #53443 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3946 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)(17,18,22,25)(19,23,20,27)(21,24,30,29)(26,31,28,32)(35,40)(37,42)(38,46)(41,44);(1,3,6,12)(2,7,9,5)(4,11,14,10)(8,13,16,15)(17,18)(19,23)(20,27)(21,24)(22,25)(26,31)(28,32)(29,30)(33,34,39,36)(35,42,40,37)(38,41,46,44)(43,48,47,45);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,16)(11,12)(17,19,22,20)(18,23,25,27)(21,26,30,28)(24,31,29,32)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,48)(46,47);(1,4)(2,8)(3,11)(5,15)(6,14)(7,13)(9,16)(10,12)(17,18)(19,23)(20,27)(21,29)(22,25)(24,30)(26,32)(28,31)(33,36,39,34)(35,42,40,37)(38,41,46,44)(43,45,47,48);(1,3)(2,5)(4,10)(6,12)(7,9)(8,13)(11,14)(15,16)(17,20,22,19)(18,23,25,27)(21,28,30,26)(24,31,29,32)(33,37,39,42)(34,40,36,35)(38,45,46,48)(41,47,44,43);(1,5)(2,3)(4,13)(6,7)(8,10)(9,12)(11,16)(14,15)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,38,39,46)(34,41,36,44)(35,43,40,47)(37,45,42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=g,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, 10329714);