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 | |
4141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,9,4,8)(5,13,14,10)(11,15,12,16)(17,18)(19,21)(20,23)(22,30)(24,26)(25,32)(27,28)(29,31)(33,34,39,38)(35,42,37,40)(36,41,44,46)(43,48,45,47);(1,3,7,4)(2,8,6,9)(5,11,14,12)(10,15,13,16)(17,19,24,23)(18,21,26,20)(22,27,32,31)(25,29,30,28)(36,44)(41,46)(43,45)(47,48);(2,6)(8,9)(10,13)(15,16)(17,20,24,21)(18,19,26,23)(22,28,32,29)(25,27,30,31)(33,35,39,37)(34,40,38,42)(36,43,44,45)(41,47,46,48);(1,4,7,3)(2,9,6,8)(5,12,14,11)(10,16,13,15)(17,21,24,20)(18,19,26,23)(22,29,32,28)(25,27,30,31)(33,36,39,44)(34,41,38,46)(35,43,37,45)(40,47,42,48);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,22,24,32)(18,25,26,30)(19,27,23,31)(20,28,21,29)(33,37,39,35)(34,42,38,40)(36,45,44,43)(41,48,46,47);(1,6,7,2)(3,9,4,8)(5,13,14,10)(11,16,12,15)(17,23,24,19)(18,20,26,21)(22,31,32,27)(25,28,30,29)(33,38,39,34)(35,42,37,40)(36,46,44,41)(43,48,45,47)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=gh,d^2=h,e^2=gh,f^2=h,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, 8092037);