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 | |
4119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,14,15,9)(5,13,16,10)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,30,32,31)(33,34)(35,40)(36,39)(37,43)(38,41)(42,47)(44,46)(45,48);(2,7)(5,16)(8,11)(9,14)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,31,32,30)(33,34,39,36)(35,40,37,43)(38,44,46,41)(42,48,45,47);(2,7)(3,12)(9,14)(10,13)(18,19)(23,25)(24,26)(30,31)(33,35,39,37)(34,40,36,43)(38,42,46,45)(41,47,44,48);(1,3,6,12)(2,8,7,11)(4,13,15,10)(5,9,16,14)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,39,34)(35,40,37,43)(38,44,46,41)(42,47,45,48);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,37,39,35)(34,40,36,43)(38,45,46,42)(41,47,44,48);(1,5,6,16)(2,10,7,13)(3,9,12,14)(4,11,15,8)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,38,39,46)(34,41,36,44)(35,42,37,45)(40,47,43,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=ghi,d^2=gi,e^2=gh,f^2=g,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, 8662363);