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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(33,34,39,41)(35,40,36,44)(37,38,47,48)(42,46,45,43);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,18)(19,24)(20,21)(22,23)(25,30)(26,32)(27,29)(28,31)(33,35,39,36)(34,40,41,44)(37,45,47,42)(38,43,48,46);(5,13)(8,10)(11,16)(14,15)(18,23)(21,25)(24,31)(27,32)(33,36,39,35)(34,40,41,44)(37,45,47,42)(38,46,48,43);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,19)(18,24)(20,29)(21,27)(22,28)(23,31)(25,32)(26,30)(33,37)(34,38)(35,42)(36,45)(39,47)(40,46)(41,48)(43,44);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20)(18,25)(19,26)(21,23)(22,30)(24,27)(28,29)(31,32)(33,38)(34,37)(35,43)(36,46)(39,48)(40,45)(41,47)(42,44);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,25)(18,20,23,30)(19,27,28,32)(24,29,31,26)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=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, 8278759);