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 | |
4139 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)(3,7)(4,8)(5,11)(6,9)(10,13)(12,15)(14,16)(17,18)(19,20)(21,24)(22,30)(23,26)(25,31)(27,29)(28,32)(33,34,39,37)(35,40,46,45)(36,42,47,41)(38,48,44,43);(1,3,6,11)(2,7,9,5)(4,10,14,12)(8,13,16,15)(17,19,23,24)(18,21,26,20)(22,27,31,32)(25,29,30,28)(33,35,39,46)(34,40,37,45)(36,44,47,38)(41,48,42,43);(1,3)(2,5)(4,10)(6,11)(7,9)(8,15)(12,14)(13,16)(17,20,23,21)(18,24,26,19)(22,28,31,29)(25,32,30,27)(33,36)(34,41)(35,44)(37,42)(38,46)(39,47)(40,48)(43,45);(1,3)(2,5)(4,12)(6,11)(7,9)(8,13)(10,14)(15,16)(17,21,23,20)(18,24,26,19)(22,29,31,28)(25,32,30,27)(33,36)(34,42)(35,44)(37,41)(38,46)(39,47)(40,43)(45,48);(1,4,6,14)(2,8,9,16)(3,10,11,12)(5,15,7,13)(17,18)(19,21)(20,24)(22,30)(23,26)(25,31)(27,28)(29,32)(33,37,39,34)(35,45,46,40)(36,42,47,41)(38,48,44,43);(1,5)(2,3)(4,13)(6,7)(8,12)(9,11)(10,16)(14,15)(17,22,23,31)(18,25,26,30)(19,27,24,32)(20,28,21,29)(33,38,39,44)(34,43,37,48)(35,36,46,47)(40,41,45,42)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=g,d^2=g,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10436997);