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 |
| prime radical | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3709 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)(6,10)(11,15)(14,16)(17,18,23,28)(19,27,22,24)(20,21,26,25)(29,32,31,30)(35,37)(39,41)(43,45)(47,48);(1,2)(3,10)(4,12)(5,8)(6,9)(7,13)(11,14)(15,16)(17,19,23,22)(18,24,28,27)(20,31,26,29)(21,32,25,30)(33,34,38,42)(35,39,37,41)(36,44,46,40)(43,48,45,47);(1,2,5,8)(3,6,9,10)(4,7,13,12)(11,14,15,16)(17,20,23,26)(18,25,28,21)(19,29,22,31)(24,32,27,30)(33,35,38,37)(34,39,42,41)(36,43,46,45)(40,47,44,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,22)(18,24,28,27)(20,29,26,31)(21,30,25,32);(1,3,5,9)(2,6,8,10)(4,11,13,15)(7,14,12,16)(17,21,23,25)(18,26,28,20)(19,30,22,32)(24,31,27,29)(33,36,38,46)(34,40,42,44)(35,43,37,45)(39,47,41,48);(1,4)(2,7)(3,11)(5,13)(6,14)(8,12)(9,15)(10,16)(17,22,23,19)(18,27,28,24)(20,31,26,29)(21,32,25,30)(33,37,38,35)(34,41,42,39)(36,45,46,43)(40,48,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ghi,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664371);