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 | |
3750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,9,10)(5,8,14,12)(11,15,13,16)(19,22)(24,26)(27,30)(31,32)(33,34,39,42)(35,43,45,38)(36,40,47,48)(37,41,46,44);(3,9)(7,10)(11,13)(15,16)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,35)(34,38)(36,44)(37,48)(39,45)(40,46)(41,47)(42,43);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,16)(13,14)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,36,39,47)(34,40,42,48)(35,44,45,41)(37,38,46,43);(1,4,6,2)(3,10,9,7)(5,12,14,8)(11,16,13,15)(17,20,23,18)(19,24,22,26)(21,28,29,25)(27,31,30,32)(33,34)(35,38)(36,40)(37,41)(39,42)(43,45)(44,46)(47,48);(1,3)(2,7)(4,10)(5,13)(6,9)(8,16)(11,14)(12,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,37)(34,41)(35,40)(36,38)(39,46)(42,44)(43,47)(45,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,22,23,19)(18,24,20,26)(21,27,29,30)(25,32,28,31)(33,38)(34,35)(36,46)(37,47)(39,43)(40,44)(41,48)(42,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=gh,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9722119);