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 | |
4134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,10,5,8)(4,9,12,14)(11,16,13,15)(18,22)(19,25)(24,27)(26,31)(33,34,40,44)(35,45,47,41)(36,48,46,39)(37,43,38,42);(4,12)(9,14)(11,13)(15,16)(18,22)(23,29)(24,27)(30,32)(33,35)(34,41)(36,37)(38,46)(39,43)(40,47)(42,48)(44,45);(1,3,7,5)(2,8,6,10)(4,11,12,13)(9,15,14,16)(17,18,21,22)(19,23,25,29)(20,27,28,24)(26,32,31,30)(33,36,40,46)(34,39,44,48)(35,37,47,38)(41,43,45,42);(1,4,7,12)(2,9,6,14)(3,11,5,13)(8,15,10,16)(33,37,40,38)(34,42,44,43)(35,36,47,46)(39,41,48,45);(1,5,7,3)(2,10,6,8)(4,13,12,11)(9,16,14,15)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32)(33,38,40,37)(34,43,44,42)(35,46,47,36)(39,45,48,41);(1,6,7,2)(3,10,5,8)(4,14,12,9)(11,16,13,15)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)(33,39)(34,36)(35,43)(37,41)(38,45)(40,48)(42,47)(44,46)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=hi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8095211);