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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3801 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,10,5,8)(4,9,12,13)(11,16,14,15)(17,18)(19,20)(21,22)(23,27)(24,25)(26,32)(28,29)(30,31)(33,34,39,44)(35,45,47,40)(36,43,38,41)(37,42,46,48);(1,3,7,5)(2,8,6,10)(4,11,12,14)(9,15,13,16)(17,19,23,25)(18,24,27,20)(21,30,26,28)(22,29,32,31)(33,34)(35,45)(36,41)(37,48)(38,43)(39,44)(40,47)(42,46);(4,12)(9,13)(11,14)(15,16)(17,20,23,24)(18,25,27,19)(21,29,26,31)(22,30,32,28)(33,35,39,47)(34,40,44,45)(36,46,38,37)(41,48,43,42);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,20)(18,19)(21,31)(22,30)(23,24)(25,27)(26,29)(28,32)(33,36,39,38)(34,41,44,43)(35,46,47,37)(40,48,45,42);(1,5,7,3)(2,8,6,10)(4,11,12,14)(9,16,13,15)(17,21)(18,22)(19,28)(20,29)(23,26)(24,31)(25,30)(27,32)(33,37)(34,42)(35,36)(38,47)(39,46)(40,41)(43,45)(44,48);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,22)(18,26)(19,29)(20,30)(21,27)(23,32)(24,28)(25,31)(33,38,39,36)(34,43,44,41)(35,37,47,46)(40,42,45,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=h,d^2=i,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9117571);