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 | |
3979 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)(19,24)(20,22)(21,30)(23,25)(26,29)(27,32)(28,31)(33,34,39,40)(35,43,44,41)(36,47,48,37)(38,45,46,42);(2,7)(5,16)(8,11)(9,14)(17,19)(18,24)(20,29)(21,27)(22,26)(23,28)(25,31)(30,32)(34,40)(35,44)(37,47)(42,45);(2,7)(3,12)(9,14)(10,13)(18,25)(19,28)(20,27)(26,30)(33,35,39,44)(34,41,40,43)(36,45,48,42)(37,46,47,38);(1,3,6,12)(2,8,7,11)(4,13,15,10)(5,9,16,14)(17,20,23,27)(18,22,25,32)(19,29,28,21)(24,26,31,30)(33,36)(34,37)(35,45)(38,43)(39,48)(40,47)(41,46)(42,44);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,21,23,29)(18,26,25,30)(19,27,28,20)(22,31,32,24)(33,37)(34,36)(35,46)(38,44)(39,47)(40,48)(41,45)(42,43);(1,5,6,16)(2,10,7,13)(3,9,12,14)(4,11,15,8)(17,22)(18,27)(19,30)(20,25)(21,31)(23,32)(24,29)(26,28)(33,38,39,46)(34,42,40,45)(35,37,44,47)(36,43,48,41)Polycyclic presentation:
2^9|a^2=g,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, 8662251);