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 | |
4131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,10,13,7)(4,5,15,9)(11,12,14,16)(17,18)(19,20)(21,29)(22,24)(23,31)(25,27)(26,30)(28,32)(33,34,40,44)(35,45,48,41)(36,42,37,47)(38,43,46,39);(1,3,6,13)(2,7,8,10)(4,14,15,11)(5,12,9,16)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,30)(31,32)(33,35)(34,41)(36,46)(37,38)(39,47)(40,48)(42,43)(44,45);(2,8)(4,15)(7,10)(11,14)(17,18,22,24)(19,25,27,20)(21,23,31,29)(26,30,32,28)(33,36,40,37)(34,42,44,47)(35,46,48,38)(39,41,43,45);(1,4)(2,9)(3,11)(5,8)(6,15)(7,12)(10,16)(13,14)(17,18)(19,25)(20,27)(21,23)(22,24)(26,30)(28,32)(29,31)(33,37,40,36)(34,42,44,47)(35,46,48,38)(39,45,43,41);(1,3)(2,10)(4,14)(5,16)(6,13)(7,8)(9,12)(11,15)(17,20,22,25)(18,19,24,27)(21,30,31,28)(23,32,29,26)(33,38,40,46)(34,43,44,39)(35,37,48,36)(41,42,45,47);(1,5,6,9)(2,4,8,15)(3,12,13,16)(7,14,10,11)(17,21)(18,23)(19,26)(20,28)(22,31)(24,29)(25,30)(27,32)(33,39,40,43)(34,38,44,46)(35,47,48,42)(36,41,37,45)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=gh,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, 10154514);