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 | |
4187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,15)(11,16)(13,14)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(34,37)(36,40)(41,45)(44,46);(1,2,6,10)(3,5,9,7)(4,8,12,15)(11,13,16,14)(20,28)(25,30)(27,29)(31,32)(33,34,39,37)(35,40,43,36)(38,41,47,45)(42,46,48,44);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,35,39,43)(34,40,37,36)(38,42,47,48)(41,46,45,44);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,16)(14,15)(17,20,23,28)(18,25,22,30)(19,27,21,29)(24,31,26,32)(33,36)(34,35)(37,43)(38,46)(39,40)(41,48)(42,45)(44,47);(1,4,6,12)(2,8,10,15)(3,11,9,16)(5,13,7,14)(17,21,23,19)(18,26,22,24)(20,29,28,27)(25,32,30,31)(33,37,39,34)(35,36,43,40)(38,45,47,41)(42,44,48,46);(1,5)(2,9)(3,10)(4,13)(6,7)(8,16)(11,15)(12,14)(17,22,23,18)(19,26,21,24)(20,30,28,25)(27,32,29,31)(33,38,39,47)(34,41,37,45)(35,42,43,48)(36,44,40,46)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=gh,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8352603);