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 | |
4281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,9,10,7)(4,8,12,13)(11,15,16,14)(17,18,22,25)(19,21,28,31)(20,24,30,23)(26,32,27,29)(33,34,39,43)(35,38,44,40)(36,45,47,41)(37,42,48,46);(2,5)(7,9)(8,13)(14,15)(17,19,22,28)(18,21,25,31)(20,27,30,26)(23,32,24,29)(33,35,39,44)(34,40,43,38)(36,37,47,48)(41,42,45,46);(3,10)(4,12)(7,9)(8,13)(17,20,22,30)(18,23,25,24)(19,26,28,27)(21,29,31,32)(33,36,39,47)(34,41,43,45)(35,37,44,48)(38,46,40,42);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,25)(26,28)(31,32)(33,36)(34,41)(35,37)(38,46)(39,47)(40,42)(43,45)(44,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,19)(18,21)(20,27)(22,28)(23,32)(24,29)(25,31)(26,30)(33,37)(34,42)(35,36)(38,45)(39,48)(40,41)(43,46)(44,47);(1,5,6,2)(3,9,10,7)(4,13,12,8)(11,15,16,14)(17,21)(18,19)(20,29)(22,31)(23,26)(24,27)(25,28)(30,32)(33,38,39,40)(34,35,43,44)(36,46,47,42)(37,45,48,41)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8669284);