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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3932 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)(3,10,4,6)(5,13,15,8)(11,14,12,16)(17,18,22,23)(19,21,26,24)(20,28,30,25)(27,31,32,29)(33,34,39,38)(35,42,37,40)(36,44,46,41)(43,47,45,48);(1,2,7,9)(3,10,4,6)(5,8,15,13)(11,16,12,14)(18,23)(19,26)(25,28)(27,32)(33,35,39,37)(34,40,38,42)(36,43,46,45)(41,47,44,48);(1,3,7,4)(2,6,9,10)(5,11,15,12)(8,14,13,16)(17,19,22,26)(18,24,23,21)(20,27,30,32)(25,31,28,29)(33,36,39,46)(34,41,38,44)(35,43,37,45)(40,47,42,48);(1,4,7,3)(2,6,9,10)(5,12,15,11)(8,14,13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,35,39,37)(34,40,38,42)(36,43,46,45)(41,47,44,48);(1,5,7,15)(2,8,9,13)(3,11,4,12)(6,14,10,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31)(33,37,39,35)(34,42,38,40)(36,45,46,43)(41,48,44,47);(1,6)(2,3)(4,9)(5,14)(7,10)(8,11)(12,13)(15,16)(17,21,22,24)(18,26,23,19)(20,29,30,31)(25,32,28,27)(33,38,39,34)(35,42,37,40)(36,41,46,44)(43,47,45,48)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=h,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8095533);