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 | |
3887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18,23,24)(19,25,20,21)(22,26,31,32)(27,30,28,29)(33,34,37,40)(35,38,41,45)(36,39,44,48)(42,46,43,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(18,24)(21,25)(26,32)(29,30)(35,41)(38,45)(42,43)(46,47);(2,6)(7,9)(8,12)(14,15)(17,19,23,20)(18,25,24,21)(22,27,31,28)(26,30,32,29)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,44)(47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,23,19)(18,21,24,25)(22,28,31,27)(26,29,32,30)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,47)(44,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21)(18,19)(20,24)(22,30)(23,25)(26,28)(27,32)(29,31)(33,35)(34,38)(36,43)(37,41)(39,47)(40,45)(42,44)(46,48);(1,4,5,13)(2,8,6,12)(3,10,11,16)(7,14,9,15)(17,22,23,31)(18,26,24,32)(19,27,20,28)(21,29,25,30)(33,36)(34,39)(35,42)(37,44)(38,46)(40,48)(41,43)(45,47)Polycyclic presentation:
2^9|a^2=ghi,c^2=gh,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820039);