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 | |
3983 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34)(35,43)(36,41)(37,42)(38,39)(40,46)(44,48)(45,47);(2,6)(7,9)(8,12)(14,15)(18,19)(23,25)(24,26)(30,31)(34,39)(40,43)(41,42)(47,48);(3,10)(7,9)(11,16)(14,15)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,46)(44,47)(45,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35,38,46)(34,40,39,43)(36,44,37,45)(41,47,42,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36,38,37)(34,41,39,42)(35,44,46,45)(40,47,43,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37,38,36)(34,42,39,41)(35,45,46,44)(40,48,43,47)Polycyclic presentation:
2^9|c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8021564);