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 | |
4091 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)(3,7)(4,9)(5,10)(6,11)(8,12)(13,15)(14,16)(17,18,22,21)(19,24,27,23)(20,28,29,25)(26,30,32,31)(33,34,38,42)(35,39,45,48)(36,46,44,40)(37,41,43,47);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,15)(10,16)(17,19,22,27)(18,23,21,24)(20,26,29,32)(25,30,28,31)(33,35,38,45)(34,39,42,48)(36,37,44,43)(40,47,46,41);(3,12)(7,8)(13,16)(14,15)(17,19,22,27)(18,23,21,24)(20,26,29,32)(25,30,28,31)(33,36,38,44)(34,40,42,46)(35,43,45,37)(39,41,48,47);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,30)(29,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,41)(42,46)(47,48);(1,4)(2,9)(3,13)(5,11)(6,10)(7,15)(8,14)(12,16)(17,20,22,29)(18,25,21,28)(19,26,27,32)(23,30,24,31)(33,34)(35,39)(36,40)(37,47)(38,42)(41,43)(44,46)(45,48);(1,5,6,9)(2,10,11,4)(3,14,12,15)(7,16,8,13)(17,21,22,18)(19,24,27,23)(20,25,29,28)(26,30,32,31)(33,37,38,43)(34,41,42,47)(35,44,45,36)(39,40,48,46)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8653509);