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 | |
4027 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,10,13,7)(4,9,14,15)(5,12,16,11)(17,18,22,23)(19,21,25,30)(20,24,28,31)(26,29,32,27)(33,34)(35,37)(36,41)(38,42)(39,40)(43,45)(44,46)(47,48);(1,3)(2,7)(4,12)(5,9)(6,13)(8,10)(11,14)(15,16)(17,19)(18,21)(20,26)(22,25)(23,30)(24,29)(27,31)(28,32)(34,40)(37,41)(42,48)(45,46);(2,8)(4,14)(7,10)(11,12)(18,23)(19,25)(24,31)(26,32)(33,35,39,36)(34,37,40,41)(38,43,47,44)(42,45,48,46);(1,4)(2,9)(3,12)(5,7)(6,14)(8,15)(10,16)(11,13)(17,18)(19,21)(20,24)(22,23)(25,30)(26,29)(27,32)(28,31)(33,36,39,35)(34,41,40,37)(38,44,47,43)(42,46,48,45);(1,3)(2,10)(4,12)(5,15)(6,13)(7,8)(9,16)(11,14)(17,20,22,28)(18,24,23,31)(19,26,25,32)(21,29,30,27)(33,37,39,41)(34,35,40,36)(38,46,47,45)(42,44,48,43);(1,5,6,16)(2,11,8,12)(3,9,13,15)(4,7,14,10)(17,21)(18,19)(20,27)(22,30)(23,25)(24,32)(26,31)(28,29)(33,38,39,47)(34,42,40,48)(35,43,36,44)(37,45,41,46)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,d^2=ghi,e^2=gh,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, 9830843);