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 | |
4161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32)(34,36)(39,42)(40,44)(46,48);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,18,23,22)(19,26,20,24)(21,25,30,29)(27,32,28,31)(35,41)(39,42)(43,47)(46,48);(6,14)(9,12)(11,13)(15,16)(17,19,23,20)(18,24,22,26)(21,27,30,28)(25,31,29,32)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(17,20,23,19)(18,24,22,26)(21,28,30,27)(25,31,29,32)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,21,23,30)(18,25,22,29)(19,27,20,28)(24,31,26,32)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,6,7,14)(2,9,4,12)(3,11,5,13)(8,15,10,16)(17,22,23,18)(19,24,20,26)(21,29,30,25)(27,31,28,32)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=h,d^2=gh,e^2=gh,f^2=gh,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, 8092022);