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 | |
4025 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)(3,10)(4,9)(5,14)(6,8)(7,13)(11,12)(15,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34,38,36)(35,39,43,41)(37,40,45,44)(42,46,48,47);(1,3)(2,7)(4,11)(5,12)(6,13)(8,10)(9,15)(14,16)(19,27)(21,30)(23,32)(25,31)(34,36)(39,41)(40,44)(46,47);(2,8)(4,14)(7,10)(11,16)(17,19,22,27)(18,23,26,32)(20,21,28,30)(24,31,29,25)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,4)(2,9)(3,11)(5,8)(6,14)(7,15)(10,12)(13,16)(17,20,22,28)(18,24,26,29)(19,21,27,30)(23,31,32,25);(1,3)(2,10)(4,11)(5,15)(6,13)(7,8)(9,12)(14,16)(17,21)(18,25)(19,28)(20,27)(22,30)(23,24)(26,31)(29,32)(33,36,38,34)(35,41,43,39)(37,40,45,44)(42,46,48,47);(1,5,6,9)(2,4,8,14)(3,12,13,15)(7,11,10,16)(17,18)(19,23)(20,24)(21,31)(22,26)(25,30)(27,32)(28,29)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gh,c^2=ghi,d^2=i,e^2=gh,f^2=h,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, 9831196);