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 | |
4134 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)(3,9)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(18,19)(23,25)(24,26)(30,31)(33,34,39,41)(35,40,37,38)(36,42,46,48)(43,45,44,47);(2,4)(3,10)(8,12)(11,13)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30)(33,35)(34,40)(36,44)(37,39)(38,41)(42,47)(43,46)(45,48);(2,4)(7,9)(8,12)(15,16)(20,27)(23,25)(28,32)(30,31)(34,41)(35,37)(42,48)(43,44);(1,3,6,10)(2,7,4,9)(5,13,14,11)(8,16,12,15)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,36)(34,42)(35,43)(37,44)(38,47)(39,46)(40,45)(41,48);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,37,39,35)(34,40,41,38)(36,44,46,43)(42,45,48,47);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,40)(34,35,41,37)(36,45,46,47)(42,44,48,43)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8529195);