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 | |
3975 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)(3,7)(4,9)(5,13)(6,8)(10,15)(11,12)(14,16)(17,18,23,26)(19,20,21,24)(22,30,31,25)(27,32,29,28)(33,34,40,44)(35,45,48,41)(36,42,37,47)(38,43,46,39);(1,3,6,9)(2,7,8,4)(5,11,15,16)(10,14,13,12)(17,19)(18,24)(20,26)(21,23)(22,27)(25,32)(28,30)(29,31)(33,35)(34,41)(36,46)(37,38)(39,47)(40,48)(42,43)(44,45);(2,8)(3,9)(10,13)(11,16)(17,20,23,24)(18,19,26,21)(22,28,31,32)(25,27,30,29)(33,36,40,37)(34,42,44,47)(35,46,48,38)(39,41,43,45);(1,2)(3,4)(5,10)(6,8)(7,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,30)(31,32)(33,37,40,36)(34,42,44,47)(35,46,48,38)(39,45,43,41);(1,4)(2,9)(3,8)(5,14)(6,7)(10,11)(12,15)(13,16)(17,21,23,19)(18,24,26,20)(22,27,31,29)(25,28,30,32)(33,38,40,46)(34,43,44,39)(35,37,48,36)(41,42,45,47);(1,5,6,15)(2,10,8,13)(3,11,9,16)(4,12,7,14)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(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=i,c^2=g,d^2=ghi,e^2=g,f^2=gh,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, 10154876);