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 | |
4012 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)(3,8)(4,6)(5,9)(7,11)(10,16)(12,14)(13,15)(17,18,23,25)(19,20,27,21)(22,24,31,32)(26,28,30,29)(34,38)(36,43)(39,44)(42,47);(1,3)(2,7)(4,8)(5,10)(6,11)(9,15)(12,16)(13,14)(17,19)(18,20)(21,25)(22,30)(23,27)(24,29)(26,31)(28,32)(35,41)(39,44)(42,47)(46,48);(17,20,23,21)(18,19,25,27)(22,28,31,29)(24,26,32,30)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,4,6,2)(3,8,11,7)(5,9,14,12)(10,15,13,16)(17,21,23,20)(18,19,25,27)(22,28,31,29)(24,30,32,26);(1,3)(2,8)(4,7)(5,13)(6,11)(9,15)(10,14)(12,16)(17,22)(18,24)(19,26)(20,28)(21,29)(23,31)(25,32)(27,30)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,5,6,14)(2,9,4,12)(3,10,11,13)(7,15,8,16)(17,19)(18,21)(20,25)(22,30)(23,27)(24,28)(26,31)(29,32)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9131501);