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 | |
4299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,9)(10,14)(11,16)(33,34,38,39)(35,43,46,40)(36,41,47,45)(37,44,48,42);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18,22,23)(19,24,27,32)(20,25,30,29)(21,26,31,28)(34,39)(36,47)(40,43)(42,44);(3,9)(4,8)(5,13)(10,14)(18,23)(19,27)(20,30)(21,31)(33,35,38,46)(34,40,39,43)(36,42,47,44)(37,41,48,45);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,19)(18,24)(20,26)(21,29)(22,27)(23,32)(25,31)(28,30)(33,35)(34,40)(36,42)(37,45)(38,46)(39,43)(41,48)(44,47);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,20)(18,25)(19,28)(21,24)(22,30)(23,29)(26,27)(31,32)(33,36)(34,41)(35,44)(37,40)(38,47)(39,45)(42,46)(43,48);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,21)(18,26)(19,29)(20,24)(22,31)(23,28)(25,27)(30,32)(33,37)(34,42)(35,45)(36,40)(38,48)(39,44)(41,46)(43,47)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,e^2=gh,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, 8884512);