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 | #2733 | C22 |
| 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)(3,12)(4,10)(5,11)(6,8)(7,9)(13,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,32)(28,31)(29,30)(33,34)(35,45)(36,41)(37,42)(38,43)(39,44)(40,46)(47,48);(1,3,7,6)(2,8,9,12)(4,13,14,11)(5,15,16,10)(17,19)(18,22)(20,26)(21,29)(23,25)(24,30)(27,31)(28,32)(33,35,39,46)(34,40,44,45)(36,37,43,47)(38,42,41,48);(2,9)(4,14)(8,12)(11,13)(18,20)(19,25)(24,28)(27,29)(33,36,39,43)(34,41,44,38)(35,37,46,47)(40,48,45,42);(1,4,7,14)(2,10,9,15)(3,13,6,11)(5,12,16,8)(17,20,23,18)(19,26,25,22)(21,28,31,24)(27,30,29,32);(1,5)(2,11)(3,10)(4,12)(6,15)(7,16)(8,14)(9,13)(17,21,23,31)(18,24,20,28)(19,27,25,29)(22,32,26,30)(33,37)(34,42)(35,43)(36,46)(38,45)(39,47)(40,41)(44,48);(1,6,7,3)(2,12,9,8)(4,11,14,13)(5,10,16,15)(17,22,23,26)(18,25,20,19)(21,30,31,32)(24,27,28,29)(33,38,39,41)(34,43,44,36)(35,42,46,48)(37,40,47,45)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,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, 9620939);