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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4016 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)(3,8,6,16)(4,10,13,5)(11,12,14,15)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,32)(30,31)(33,34,39,37)(35,40,44,47)(36,45,46,41)(38,48,43,42);(1,3,7,6)(2,8,9,16)(4,11,13,14)(5,15,10,12)(17,19,23,22)(18,24,27,28)(20,29,21,30)(25,31,26,32)(33,35,39,44)(34,40,37,47)(36,43,46,38)(41,48,45,42);(2,9)(4,13)(8,16)(11,14)(17,20,23,21)(18,25,27,26)(19,29,22,30)(24,31,28,32)(33,36,39,46)(34,41,37,45)(35,38,44,43)(40,42,47,48);(1,4)(2,10)(3,11)(5,9)(6,14)(7,13)(8,12)(15,16)(17,21,23,20)(18,26,27,25)(19,30,22,29)(24,32,28,31)(33,36)(34,41)(35,43)(37,45)(38,44)(39,46)(40,48)(42,47);(1,5,7,10)(2,4,9,13)(3,12,6,15)(8,14,16,11)(17,22,23,19)(18,24,27,28)(20,30,21,29)(25,31,26,32)(33,37,39,34)(35,40,44,47)(36,45,46,41)(38,42,43,48);(1,6,7,3)(2,8,9,16)(4,14,13,11)(5,15,10,12)(17,18)(19,24)(20,25)(21,26)(22,28)(23,27)(29,31)(30,32)(33,38,39,43)(34,42,37,48)(35,36,44,46)(40,41,47,45)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=gh,f^2=h,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, 9630366);