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 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3831 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,11)(3,7,14,16)(4,9,13,8)(5,10,12,15)(34,40)(35,45)(36,38)(43,48);(1,3,6,14)(2,7,11,16)(4,12,13,5)(8,10,9,15)(17,18,22,26)(19,24,29,23)(20,21,25,32)(27,31,30,28)(33,34)(35,46)(36,42)(37,48)(38,44)(39,40)(41,45)(43,47);(1,4,6,13)(2,8,11,9)(3,5,14,12)(7,15,16,10)(17,19,22,29)(18,23,26,24)(20,30,25,27)(21,31,32,28)(33,35)(34,41)(36,37)(38,47)(39,45)(40,46)(42,43)(44,48);(1,4)(2,9)(3,12)(5,14)(6,13)(7,15)(8,11)(10,16)(17,20,22,25)(18,21,26,32)(19,27,29,30)(23,28,24,31)(33,36,39,38)(34,42,40,44)(35,37,45,47)(41,43,46,48);(1,5,6,12)(2,10,11,15)(3,13,14,4)(7,8,16,9)(17,19)(18,24)(20,30)(21,28)(22,29)(23,26)(25,27)(31,32)(33,37,39,47)(34,43,40,48)(35,36,45,38)(41,42,46,44);(1,2)(3,7)(4,9)(5,10)(6,11)(8,13)(12,15)(14,16)(17,21)(18,25)(19,28)(20,26)(22,32)(23,30)(24,27)(29,31)(33,38,39,36)(34,44,40,42)(35,47,45,37)(41,48,46,43)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8901915);