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 | |
3805 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,32)(28,31)(29,30)(33,34,39,43)(35,42,38,40)(36,37,41,48)(44,45,46,47);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19)(18,22)(20,26)(21,29)(23,25)(24,30)(27,31)(28,32)(33,35,39,38)(34,40,43,42)(36,44,41,46)(37,47,48,45);(2,9)(3,4)(10,16)(11,12)(18,20)(19,25)(24,28)(27,29)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,48,47);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,20,23,18)(19,26,25,22)(21,28,31,24)(27,30,29,32)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,47,42,45);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,21,23,31)(18,24,20,28)(19,27,25,29)(22,32,26,30)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,44)(42,46);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,22,23,26)(18,25,20,19)(21,30,31,32)(24,27,28,29)(33,38,39,35)(34,42,43,40)(36,46,41,44)(37,45,48,47)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=hi,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9620903);