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 | |
3975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,11,16)(4,6,14,15)(5,13,12,10)(17,18,22,23)(19,26,29,24)(20,21,30,25)(27,31,32,28)(33,34,39,36)(35,38,43,40)(37,41,46,44)(42,45,48,47);(1,3,7,11)(2,8,9,16)(4,12,14,5)(6,10,15,13)(18,23)(21,25)(24,26)(28,31)(33,34,39,36)(35,40,43,38)(37,44,46,41)(42,45,48,47);(2,9)(3,11)(4,14)(10,13)(17,19,22,29)(18,24,23,26)(20,27,30,32)(21,28,25,31)(33,35,39,43)(34,40,36,38)(37,42,46,48)(41,47,44,45);(1,4)(2,6)(3,12)(5,11)(7,14)(8,10)(9,15)(13,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,31)(30,32)(33,36,39,34)(35,38,43,40)(37,44,46,41)(42,45,48,47);(1,5,7,12)(2,10,9,13)(3,4,11,14)(6,8,15,16)(17,20)(18,25)(19,27)(21,23)(22,30)(24,31)(26,28)(29,32)(33,37,39,46)(34,41,36,44)(35,42,43,48)(38,45,40,47);(1,6)(2,4)(3,13)(5,8)(7,15)(9,14)(10,11)(12,16)(17,21,22,25)(18,20,23,30)(19,28,29,31)(24,27,26,32)(33,38)(34,35)(36,43)(37,45)(39,40)(41,42)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=hi,d^2=ghi,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, 8662647);