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 | |
4237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,12,13,8)(4,14,11,6)(5,10,15,16)(17,18)(19,27)(20,23)(21,25)(22,31)(24,29)(26,32)(28,30)(33,34,39,36)(35,42,37,40)(38,44,46,41)(43,47,45,48);(1,3)(2,8)(4,5)(6,10)(7,13)(9,12)(11,15)(14,16)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,26,30,31)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(2,9)(3,13)(4,11)(10,16)(18,20)(24,27)(25,30)(26,31)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,4,7,11)(2,6,9,14)(3,5,13,15)(8,10,12,16)(17,20,23,18)(19,27,29,24)(21,30,28,25)(22,31,32,26)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,5)(2,10)(3,11)(4,13)(6,12)(7,15)(8,14)(9,16)(17,21)(18,25)(19,22)(20,30)(23,28)(24,26)(27,31)(29,32)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,6)(2,11)(3,10)(4,9)(5,12)(7,14)(8,15)(13,16)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,30)(25,27)(33,38,39,46)(34,41,36,44)(35,43,37,45)(40,47,42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9701556);