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 | |
4074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,11,5)(4,6,9,15)(12,13,16,14)(19,25)(22,26)(27,31)(30,32)(34,37)(35,43)(36,44)(42,47);(1,3)(2,8)(4,12)(5,10)(6,13)(7,11)(9,16)(14,15)(17,18,21,24)(19,26,25,22)(20,23,29,28)(27,32,31,30)(33,34)(35,40)(36,45)(37,39)(38,42)(41,44)(43,46)(47,48);(3,11)(5,8)(12,16)(13,14)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,35,39,43)(34,40,37,46)(36,38,44,48)(41,47,45,42);(1,4)(2,6)(3,12)(5,14)(7,9)(8,13)(10,15)(11,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,32)(30,31)(33,36)(34,41)(35,38)(37,45)(39,44)(40,47)(42,46)(43,48);(1,5)(2,3)(4,14)(6,12)(7,8)(9,13)(10,11)(15,16)(17,19)(18,22)(20,27)(21,25)(23,30)(24,26)(28,32)(29,31)(33,37,39,34)(35,40,43,46)(36,45,44,41)(38,47,48,42);(1,6)(2,9)(3,13)(4,10)(5,12)(7,15)(8,16)(11,14)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,38)(34,42)(35,44)(36,43)(37,47)(39,48)(40,41)(45,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,e^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, 8494822);