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 | |
4174 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,22)(19,26,21,24)(20,28,30,25)(27,31,29,32)(33,34)(35,39)(36,41)(37,42)(38,40)(43,46)(44,48)(45,47);(4,11)(8,14)(10,15)(12,16)(17,19,23,21)(18,24,22,26)(20,29,30,27)(25,32,28,31)(33,35,38,42)(34,39,40,37)(36,44,47,43)(41,48,45,46);(2,7)(3,9)(8,14)(10,15)(34,40)(35,42)(41,45)(43,44);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,30)(18,25,22,28)(19,27,21,29)(24,31,26,32)(33,34)(35,37)(36,45)(38,40)(39,42)(41,47)(43,46)(44,48);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,21,23,19)(18,24,22,26)(20,27,30,29)(25,32,28,31)(33,36,38,47)(34,41,40,45)(35,43,42,44)(37,48,39,46);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22,23,18)(19,26,21,24)(20,28,30,25)(27,32,29,31)(33,37)(34,35)(36,46)(38,39)(40,42)(41,44)(43,45)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154291);