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 | #46783 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4029 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,8,13,14)(11,16,12,15)(17,18)(19,24)(20,25)(21,23)(22,26)(27,31)(28,32)(29,30)(33,34,39,42)(35,40,43,37)(36,38,41,47)(44,45,48,46);(2,6)(3,10)(8,14)(11,12)(17,19,23,27)(18,24,21,31)(20,28,30,26)(22,25,32,29)(33,35)(34,40)(36,44)(37,42)(38,45)(39,43)(41,48)(46,47);(2,6)(7,9)(8,14)(15,16)(18,21)(19,27)(25,29)(26,28)(35,43)(37,40)(44,48)(45,46);(1,3,5,10)(2,7,6,9)(4,12,13,11)(8,16,14,15)(17,20,23,30)(18,25,21,29)(19,26,27,28)(22,31,32,24)(33,36)(34,38)(35,44)(37,46)(39,41)(40,45)(42,47)(43,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,21,23,18)(19,24,27,31)(20,29,30,25)(22,28,32,26)(33,37)(34,35)(36,46)(38,44)(39,40)(41,45)(42,43)(47,48);(1,4,5,13)(2,8,6,14)(3,11,10,12)(7,15,9,16)(17,22)(18,26)(19,25)(20,24)(21,28)(23,32)(27,29)(30,31)(33,38)(34,41)(35,45)(36,42)(37,44)(39,47)(40,48)(43,46)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,d^2=h,e^2=gh,f^2=g,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, 8496068);