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 | |
4078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(6,12)(10,15)(13,16)(17,18)(19,28)(20,21)(22,25)(23,24)(26,27)(29,32)(30,31)(34,37)(35,43)(36,46)(42,48);(1,2,5,8)(3,6,9,12)(4,7,11,14)(10,13,15,16)(18,24)(20,27)(25,28)(29,31)(33,34,39,37)(35,44,43,40)(36,47,46,41)(38,42,45,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,23,22)(18,25,24,28)(20,31,27,29)(21,32,26,30)(33,35)(34,40)(36,45)(37,44)(38,46)(39,43)(41,48)(42,47);(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,20,23,27)(18,26,24,21)(19,29,22,31)(25,32,28,30)(33,36,39,46)(34,41,37,47)(35,45,43,38)(40,48,44,42);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,21,23,26)(18,27,24,20)(19,30,22,32)(25,31,28,29)(33,37,39,34)(35,44,43,40)(36,47,46,41)(38,48,45,42);(1,4,5,11)(2,7,8,14)(3,10,9,15)(6,13,12,16)(17,22,23,19)(18,28,24,25)(20,31,27,29)(21,32,26,30)(33,38,39,45)(34,42,37,48)(35,46,43,36)(40,47,44,41)Polycyclic presentation:
2^9|b^2=ghi,c^2=gi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8034804);