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 | |
4009 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,10,14,8)(4,6,5,9)(12,13,15,16)(17,18,23,21)(19,24,30,29)(20,26,31,25)(22,32,28,27)(33,34,40,37)(35,43,39,41)(36,42,47,38)(44,45,46,48);(1,3)(2,8)(4,15)(5,12)(6,13)(7,14)(9,16)(10,11)(17,19,23,30)(18,24,21,29)(20,28,31,22)(25,32,26,27)(33,35,40,39)(34,41,37,43)(36,44,47,46)(38,45,42,48);(1,4,7,5)(2,9,11,6)(3,12,14,15)(8,13,10,16)(17,20)(18,25)(19,28)(21,26)(22,30)(23,31)(24,32)(27,29)(33,36,40,47)(34,42,37,38)(35,44,39,46)(41,48,43,45);(1,5,7,4)(2,6,11,9)(3,12,14,15)(8,13,10,16)(17,20)(18,26)(19,28)(21,25)(22,30)(23,31)(24,27)(29,32)(33,37,40,34)(35,41,39,43)(36,38,47,42)(44,48,46,45);(1,3)(2,10)(4,12)(5,15)(6,13)(7,14)(8,11)(9,16)(17,21,23,18)(19,29,30,24)(20,26,31,25)(22,32,28,27)(33,38)(34,36)(35,45)(37,47)(39,48)(40,42)(41,44)(43,46);(1,6)(2,4)(3,13)(5,11)(7,9)(8,15)(10,12)(14,16)(17,22,23,28)(18,27,21,32)(19,20,30,31)(24,25,29,26)(33,39,40,35)(34,41,37,43)(36,46,47,44)(38,45,42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=g,e^2=hi,f^2=gh,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, 10413475);