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 | |
4122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,7,11,8)(4,5,9,15)(12,13,16,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31)(33,34)(35,38)(36,41)(37,42)(39,43)(40,44)(45,48)(46,47);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,16)(14,15)(17,18,21,24)(19,22,27,25)(20,23,30,28)(26,31,29,32)(33,35)(34,36)(37,39)(38,44)(40,41)(42,45)(43,46)(47,48);(3,11)(7,8)(12,16)(13,14)(17,18,21,24)(19,22,27,25)(20,23,30,28)(26,31,29,32)(33,36,40,38)(34,41,44,35)(37,43,47,45)(39,46,48,42);(1,3)(2,8)(4,12)(5,14)(6,11)(7,10)(9,16)(13,15)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,37,40,47)(34,42,44,46)(35,39,41,48)(36,45,38,43);(1,4)(2,5)(3,12)(6,9)(7,13)(8,14)(10,15)(11,16)(17,19)(18,22)(20,29)(21,27)(23,32)(24,25)(26,30)(28,31)(33,38,40,36)(34,35,44,41)(37,43,47,45)(39,46,48,42);(1,5)(2,9)(3,13)(4,10)(6,15)(7,16)(8,12)(11,14)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32)(33,39)(34,43)(35,37)(36,46)(38,42)(40,48)(41,47)(44,45)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=g,d^2=gh,e^2=gh,f^2=h,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, 8640865);